Theorem pm4.72

Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of WhiteheadRussell p. 121. (Contributed by NM, 30-Aug-1993) (Proof shortened by Wolf Lammen, 30-Jan-2013)

	Ref	Expression
Assertion	pm4.72	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		olc	$$⊢\psi \to \left(\phi \vee \psi \right)$$
2		pm2.621	$$⊢\left(\phi \to \psi \right)\to \left(\left(\phi \vee \psi \right)\to \psi \right)$$
3	1 2	impbid2	$$⊢\left(\phi \to \psi \right)\to \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$
4		orc	$$⊢\phi \to \left(\phi \vee \psi \right)$$
5		biimpr	$$⊢\left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)\to \left(\left(\phi \vee \psi \right)\to \psi \right)$$
6	4 5	syl5	$$⊢\left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)\to \left(\phi \to \psi \right)$$
7	3 6	impbii	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$