Theorem pm4.72 703

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 290	. . 3 |- (ps -> (ph \/ ps))
2		pm2.621 269	. . 3 |- ((ph -> ps) -> ((ph \/ ps) -> ps))
3	1, 2	impbid2 576	. 2 |- ((ph -> ps) -> (ps <-> (ph \/ ps)))
4		bi2 166	. . 3 |- ((ps <-> (ph \/ ps)) -> ((ph \/ ps) -> ps))
5		pm2.67 304	. . 3 |- (((ph \/ ps) -> ps) -> (ph -> ps))
6	4, 5	syl 12	. 2 |- ((ps <-> (ph \/ ps)) -> (ph -> ps))
7	3, 6	impbii 174	1 |- ((ph -> ps) <-> (ps <-> (ph \/ ps)))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239

This theorem is referenced by:  pm5.55 739  bigolden 819  ssequn1 2775  ssequn1OLD 2776  icounlem 7581  pleval2 16785  elpaddn0 17261

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 164  df-or 241  df-an 242

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