Theorem pm4.71 481

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120.

Assertion

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

Proof of Theorem pm4.71

Step	Hyp	Ref	Expression
1		ancl 242	. . 3 |- ((ph -> ps) -> (ph -> (ph /\ ps)))
2		pm3.26 256	. . . 4 |- ((ph /\ ps) -> ph)
3	2	a1i 7	. . 3 |- ((ph -> ps) -> ((ph /\ ps) -> ph))
4	1, 3	impbid 397	. 2 |- ((ph -> ps) -> (ph <-> (ph /\ ps)))
5		bi1 130	. . 3 |- ((ph <-> (ph /\ ps)) -> (ph -> (ph /\ ps)))
6		pm3.27 260	. . 3 |- ((ph /\ ps) -> ps)
7	5, 6	syl6 23	. 2 |- ((ph <-> (ph /\ ps)) -> (ph -> ps))
8	4, 7	impbi 139	1 |- ((ph -> ps) <-> (ph <-> (ph /\ ps)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  pm4.71r 482  pm4.71i 483  bigolden 513  rabid2 1309  dfss2 1497  disj3 1736  moabex 1868  dmopab2 2541  fcoi2 2766  fcnvres 2768  pw2en 3348

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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