Theorem pm4.71r 382

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		pm4.71 381	. 2 |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) )
2		ancom 262	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
3	2	bibi2i 225	. 2 |- ( ( ph <-> ( ph /\ ps ) ) <-> ( ph <-> ( ps /\ ph ) ) )
4	1, 3	bitri 182	1 |- ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm4.71ri  384  pm4.71rd  386