Theorem pm4.71r 387

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

Assertion

Ref	Expression
pm4.71r	|- ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) )

Proof of Theorem pm4.71r

Step	Hyp	Ref	Expression
1		pm4.71 386	. 2 |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) )
2		ancom 264	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
3	2	bibi2i 226	. 2 |- ( ( ph <-> ( ph /\ ps ) ) <-> ( ph <-> ( ps /\ ph ) ) )
4	1, 3	bitri 183	1 |- ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm4.71ri  389  pm4.71rd  391