Theorem pm4.45im 585

Description: Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)

Assertion

Ref	Expression
pm4.45im	|- ( ph <-> ( ph /\ ( ps -> ph ) ) )

Proof of Theorem pm4.45im

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 |- ( ph -> ( ps -> ph ) )
2	1	ancli 574	. 2 |- ( ph -> ( ph /\ ( ps -> ph ) ) )
3		simpl 473	. 2 |- ( ( ph /\ ( ps -> ph ) ) -> ph )
4	2, 3	impbii 199	1 |- ( ph <-> ( ph /\ ( ps -> ph ) ) )

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:  difdif  3736