Theorem pm4.71 611

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 443	. . 3 |- ((ph /\ ps) -> ph)
2	1	biantru 491	. 2 |- ((ph -> (ph /\ ps)) <-> ((ph -> (ph /\ ps)) /\ ((ph /\ ps) -> ph)))
3		anclb 530	. 2 |- ((ph -> ps) <-> (ph -> (ph /\ ps)))
4		dfbi2 609	. 2 |- ((ph <-> (ph /\ ps)) <-> ((ph -> (ph /\ ps)) /\ ((ph /\ ps) -> ph)))
5	2, 3, 4	3bitr4i 268	1 |- ((ph -> ps) <-> (ph <-> (ph /\ ps)))

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:  pm4.71r  612  pm4.71i  613  pm4.71d  615  bigolden  901  pm5.75  903  exintrbi  1613  rabid2  2789  dfss2  3263  disj3  3596  dmopab3  4918  resopab2  5002  resoprab2  5583