Theorem pm5.6

Description: Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of WhiteheadRussell p. 125. (Contributed by NM, 8-Jun-1994)

	Ref	Expression
Assertion	pm5.6	$$⊢\left(\left(\phi \wedge \neg \psi \right)\to \chi \right)\leftrightarrow \left(\phi \to \left(\psi \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		impexp	$$⊢\left(\left(\phi \wedge \neg \psi \right)\to \chi \right)\leftrightarrow \left(\phi \to \left(\neg \psi \to \chi \right)\right)$$
2		df-or	$$⊢\left(\psi \vee \chi \right)\leftrightarrow \left(\neg \psi \to \chi \right)$$
3	2	imbi2i	$$⊢\left(\phi \to \left(\psi \vee \chi \right)\right)\leftrightarrow \left(\phi \to \left(\neg \psi \to \chi \right)\right)$$
4	1 3	bitr4i	$$⊢\left(\left(\phi \wedge \neg \psi \right)\to \chi \right)\leftrightarrow \left(\phi \to \left(\psi \vee \chi \right)\right)$$