Theorem pm4.71 662

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)

Assertion

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

Proof of Theorem pm4.71

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( ph /\ ps ) -> ph )
2	1	biantru 526	. 2 |- ( ( ph -> ( ph /\ ps ) ) <-> ( ( ph -> ( ph /\ ps ) ) /\ ( ( ph /\ ps ) -> ph ) ) )
3		anclb 570	. 2 |- ( ( ph -> ps ) <-> ( ph -> ( ph /\ ps ) ) )
4		dfbi2 660	. 2 |- ( ( ph <-> ( ph /\ ps ) ) <-> ( ( ph -> ( ph /\ ps ) ) /\ ( ( ph /\ ps ) -> ph ) ) )
5	2, 3, 4	3bitr4i 292	1 |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  pm4.71r  663  pm4.71i  664  pm4.71d  666  bigolden  976  pm5.75  978  pm5.75OLD  979  rabid2  3118  rabid2f  3119  dfss2  3591  disj3  4021  dmopab3  5337  mptfnf  6015  nanorxor  38504