Theorem pm4.72 485

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 224	. . . 4 |- (ps -> (ph \/ ps))
2	1	a1i 7	. . 3 |- ((ph -> ps) -> (ps -> (ph \/ ps)))
3		pm2.621 211	. . 3 |- ((ph -> ps) -> ((ph \/ ps) -> ps))
4	2, 3	impbid 397	. 2 |- ((ph -> ps) -> (ps <-> (ph \/ ps)))
5		bi2 131	. . 3 |- ((ps <-> (ph \/ ps)) -> ((ph \/ ps) -> ps))
6		pm2.67 231	. . 3 |- (((ph \/ ps) -> ps) -> (ph -> ps))
7	5, 6	syl 12	. 2 |- ((ps <-> (ph \/ ps)) -> (ph -> ps))
8	4, 7	impbi 139	1 |- ((ph -> ps) <-> (ps <-> (ph \/ ps)))

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195

This theorem is referenced by:  bigolden 513  ssequn1 1628

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-or 197  df-an 198

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