Theorem pm4.71 389

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-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 109	. . 3 |- ( ( ph /\ ps ) -> ph )
2	1	biantru 302	. 2 |- ( ( ph -> ( ph /\ ps ) ) <-> ( ( ph -> ( ph /\ ps ) ) /\ ( ( ph /\ ps ) -> ph ) ) )
3		anclb 319	. 2 |- ( ( ph -> ps ) <-> ( ph -> ( ph /\ ps ) ) )
4		dfbi2 388	. 2 |- ( ( ph <-> ( ph /\ ps ) ) <-> ( ( ph -> ( ph /\ ps ) ) /\ ( ( ph /\ ps ) -> ph ) ) )
5	2, 3, 4	3bitr4i 212	1 |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm4.71r  390  pm4.71i  391  pm4.71d  393  bigolden  955  pm5.75  962  exintrbi  1631  rabid2  2651  dfss2  3142  disj3  3473  dmopab3  4833