EQ 5 Implication Axioms and Theorems

(1) Axiom: Definition of Implication: p $\Rightarrow$ q $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ p $\vee$ q $\equiv$ q

(2) Axiom: Consequence p $\Leftarrow$ q $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ q $\Rightarrow$ p

(3) Theorem: Definition of Implication p $\Rightarrow$ q $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ $\neg$p $\vee$ q

(4) Theorem: Definition of Implication: p $\Rightarrow$ q $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ p $\wedge$ q $\equiv$ p

(5) Theorem: Contrapositive p $\Rightarrow$ q $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ $\neg$q $\Rightarrow$ $\neg$ p

(6) Theorem: p $\Rightarrow$ (q $\equiv$ r)$\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ p $\wedge$ q $\equiv$ p $\wedge$ r

(7) Theorem: Distributivity $\frac{\Rightarrow}{\equiv}$ p $\Rightarrow$ (q $\equiv$ r) $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ p $\Rightarrow$ q $\equiv$ p $\Rightarrow$ r

(8) Theorem: Distributivity $\frac{\Rightarrow}{\Rightarrow}$ p $\Rightarrow$ (q $\Rightarrow$ r) $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ (p $\Rightarrow$ q) $\Rightarrow$ (p $\Rightarrow$ r)

(9) Theorem: Shunting p $\wedge$ q $\Rightarrow$ r $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ p $\Rightarrow$ (q $\Rightarrow$ r )

(10) Theorem: p $\wedge$ ( p $\Rightarrow$ q ) $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ p $\wedge$ q

(11) Theorem: p $\wedge$ ( q $\Rightarrow$ p) $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ p

(12) Theorem: p $\vee$ ( p $\Rightarrow$ q ) $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ true

(13) Theorem: p $\vee$ ( q $\Rightarrow$ p) $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ q $\Rightarrow$ p

(14) Theorem: p $\vee$ q $\hspace{0.2 cm }$$\Rightarrow$$\hspace{0.2 cm }$ p $\wedge$ q $\equiv$ p $\equiv$ q

(15) Reflexivity of $\Rightarrow$ : p $\Rightarrow$ p $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ true

(16) Theorem: Right Zero of $\Rightarrow$ : p $\Rightarrow$ true $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ true

(17) Theorem: Left Identity of $\Rightarrow$ : true $\Rightarrow$ p $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ p

(18) Theorem: p $\Rightarrow$ false $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ $\neg$p


(19) Theorem: false $\Rightarrow$ p $\hspace{0.2 cm }$$\equiv$$\hspace{0.2 cm }$ true

Weakening and Strengthening Theorems

(20) Theorem: p $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ p $\vee$ q

(21) Theorem: p $\wedge$ q $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ p

(22) Theorem: p $\wedge$ q $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ p $\vee$ q

(23) Theorem: p $\wedge$ q $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ p $\wedge$ (q $\vee$ r)

(24) Theorem: p $\vee$ (q $\wedge$ r) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ p $\vee$ q




(25) Theorem: Modus Ponens p $\wedge$ ( p $\Rightarrow$ q) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ q

(26) Theorem : (p $\Rightarrow$ r) $\wedge$ (q $\Rightarrow$ r) $\hspace{0.2 cm }$ $\equiv$ $\hspace{0.2 cm }$ ( p $\vee$ q $\Rightarrow$ r)

(27) Theorem : (p $\Rightarrow$ r) $\wedge$ ( $\neg$p $\Rightarrow$ r) $\hspace{0.2 cm }$ $\equiv$ $\hspace{0.2 cm }$ r

(28) Theorem: Mutual Implication (p $\Rightarrow$ q) $\wedge$ (q $\Rightarrow$ p) $\hspace{0.2 cm }$ $\equiv$ $\hspace{0.2 cm }$ ( p $\equiv$ q)

(29) Theorem: Anti-Symmetry (p $\Rightarrow$ q) $\wedge$ (q $\Rightarrow$ p) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ ( p $\equiv$ q)

(30) Theorem: Transitivity (1) (p $\Rightarrow$ q) $\wedge$ (q $\Rightarrow$ r) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ (p $\Rightarrow$ r)

(31) Theorem: Transitivity (2) (p $\equiv$ q) $\wedge$ (q $\Rightarrow$ r) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ (p $\Rightarrow$ r)

(32) Theorem: Transitivity (3) (p $\Rightarrow$ q) $\wedge$ (q $\equiv$ r) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ (p $\Rightarrow$ r)

Other useful Implication Theorems

(33) Theorem: p $\Rightarrow$ ( q $\Rightarrow$ p) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ true

(34) Theorem (p $\Rightarrow$ q) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ (p $\wedge$ r) $\Rightarrow$ (q $\wedge$ r)

(35) Theorem (p $\Rightarrow$ q) $\wedge$ (r $\Rightarrow$ s) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ (p $\vee$ r) $\Rightarrow$ (q $\vee$ s)

(36) Theorem (p $\Rightarrow$ q) $\wedge$ (r $\Rightarrow$ s) $\hspace{0.2 cm }$ $\Rightarrow$ $\hspace{0.2 cm }$ (p $\wedge$ r) $\Rightarrow$ (q $\wedge$ s)

(37) Theorem a $\Rightarrow$ $\neg$b $\hspace{0.2 cm }$ $\equiv$ $\hspace{0.2 cm }$ a $\Rightarrow$ b $\equiv$ $\neg$a

(38) Theorem $\neg$a $\Rightarrow$ b $\hspace{0.2 cm }$ $\equiv$ $\hspace{0.2 cm }$ a $\Rightarrow$ b $\equiv$ b