Theorem oibabs 851

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		ioran 476	. . . 4 |- (-. (ph \/ ps) <-> (-. ph /\ -. ps))
2		pm5.21 831	. . . 4 |- ((-. ph /\ -. ps) -> (ph <-> ps))
3	1, 2	sylbi 187	. . 3 |- (-. (ph \/ ps) -> (ph <-> ps))
4		id 19	. . 3 |- ((ph <-> ps) -> (ph <-> ps))
5	3, 4	ja 153	. 2 |- (((ph \/ ps) -> (ph <-> ps)) -> (ph <-> ps))
6		ax-1 6	. 2 |- ((ph <-> ps) -> ((ph \/ ps) -> (ph <-> ps)))
7	5, 6	impbii 180	1 |- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is referenced by: (None)