Theorem pm4.72 640

Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121.

Assertion

Ref	Expression
pm4.72	|- ((ph -> ps) <-> (ps <-> (ph \/ ps)))

Proof of Theorem pm4.72

Step	Hyp	Ref	Expression
1		olc 268	. . 3 |- (ps -> (ph \/ ps))
2		pm2.621 249	. . 3 |- ((ph -> ps) -> ((ph \/ ps) -> ps))
3	1, 2	impbid2 517	. 2 |- ((ph -> ps) -> (ps <-> (ph \/ ps)))
4		bi2 149	. . 3 |- ((ps <-> (ph \/ ps)) -> ((ph \/ ps) -> ps))
5		pm2.67 282	. . 3 |- (((ph \/ ps) -> ps) -> (ph -> ps))
6	4, 5	syl 10	. 2 |- ((ps <-> (ph \/ ps)) -> (ph -> ps))
7	3, 6	impbi 157	1 |- ((ph -> ps) <-> (ps <-> (ph \/ ps)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222

This theorem is referenced by:  pm5.55 674  bigolden 746  ssequn1 2197  icounlem 6358

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

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