Theorem pm4.45im 545

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 534	. 2 |- (ph -> (ph /\ (ps -> ph)))
3		simpl 443	. 2 |- ((ph /\ (ps -> ph)) -> ph)
4	2, 3	impbii 180	1 |- (ph <-> (ph /\ (ps -> ph)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

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 177  df-an 360

This theorem is referenced by:  difdif  3393