Theorem pm4.71r

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of WhiteheadRussell p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999)

	Ref	Expression
Assertion	pm4.71r	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi \leftrightarrow \left(\psi \wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm4.71	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
2		ancom	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\psi \wedge \phi \right)$$
3	2	bibi2i	$$⊢\left(\phi \leftrightarrow \left(\phi \wedge \psi \right)\right)\leftrightarrow \left(\phi \leftrightarrow \left(\psi \wedge \phi \right)\right)$$
4	1 3	bitri	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi \leftrightarrow \left(\psi \wedge \phi \right)\right)$$