Theorem oibabs 799

Description: Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)

Assertion

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

Proof of Theorem oibabs

Step	Hyp	Ref	Expression
1		pm2.67-2 633	. . . 4 |- (((ph \/ ps) -> (ph <-> ps)) -> (ph -> (ph <-> ps)))
2	1	ibd 167	. . 3 |- (((ph \/ ps) -> (ph <-> ps)) -> (ph -> ps))
3		olc 631	. . . . 5 |- (ps -> (ph \/ ps))
4	3	imim1i 54	. . . 4 |- (((ph \/ ps) -> (ph <-> ps)) -> (ps -> (ph <-> ps)))
5		ibibr 235	. . . 4 |- ((ps -> ph) <-> (ps -> (ph <-> ps)))
6	4, 5	sylibr 137	. . 3 |- (((ph \/ ps) -> (ph <-> ps)) -> (ps -> ph))
7	2, 6	impbid 120	. 2 |- (((ph \/ ps) -> (ph <-> ps)) -> (ph <-> ps))
8		ax-1 5	. 2 |- ((ph <-> ps) -> ((ph \/ ps) -> (ph <-> ps)))
9	7, 8	impbii 117	1 |- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))

Syntax hints:   -> wi 4   <-> wb 98   \/ wo 628

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)