Theorem oibabs 703

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		pm2.67-2 702	. . . 4 |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) -> ( ph -> ( ph <-> ps ) ) )
2	1	ibd 177	. . 3 |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) -> ( ph -> ps ) )
3		olc 700	. . . . 5 |- ( ps -> ( ph \/ ps ) )
4	3	imim1i 60	. . . 4 |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) -> ( ps -> ( ph <-> ps ) ) )
5		ibibr 245	. . . 4 |- ( ( ps -> ph ) <-> ( ps -> ( ph <-> ps ) ) )
6	4, 5	sylibr 133	. . 3 |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) -> ( ps -> ph ) )
7	2, 6	impbid 128	. 2 |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) -> ( ph <-> ps ) )
8		ax-1 6	. 2 |- ( ( ph <-> ps ) -> ( ( ph \/ ps ) -> ( ph <-> ps ) ) )
9	7, 8	impbii 125	1 |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) <-> ( ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)