Theorem oibabs 653

Description: Absorption of disjunction into equivalence.

Assertion

Ref	Expression
oibabs	|- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))

Proof of Theorem oibabs

Step	Hyp	Ref	Expression
1		orc 269	. . . . 5 |- (ph -> (ph \/ ps))
2	1	imim1i 16	. . . 4 |- (((ph \/ ps) -> (ph <-> ps)) -> (ph -> (ph <-> ps)))
3	2	ibd 593	. . 3 |- (((ph \/ ps) -> (ph <-> ps)) -> (ph -> ps))
4		olc 268	. . . . 5 |- (ps -> (ph \/ ps))
5	4	imim1i 16	. . . 4 |- (((ph \/ ps) -> (ph <-> ps)) -> (ps -> (ph <-> ps)))
6		ibibr 590	. . . 4 |- ((ps -> ph) <-> (ps -> (ph <-> ps)))
7	5, 6	sylibr 200	. . 3 |- (((ph \/ ps) -> (ph <-> ps)) -> (ps -> ph))
8	3, 7	impbid 515	. 2 |- (((ph \/ ps) -> (ph <-> ps)) -> (ph <-> ps))
9		ax-1 4	. 2 |- ((ph <-> ps) -> ((ph \/ ps) -> (ph <-> ps)))
10	8, 9	impbi 157	1 |- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))

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

This theorem is referenced by:  lmsslem 7914

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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