Theorem dedlema 971

Description: Lemma for iftrue 3566. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Assertion

Ref	Expression
dedlema	|- ( ph -> ( ps <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )

Proof of Theorem dedlema

Step	Hyp	Ref	Expression
1		orc 713	. . 3 |- ( ( ps /\ ph ) -> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) )
2	1	expcom 116	. 2 |- ( ph -> ( ps -> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )
3		simpl 109	. . . 4 |- ( ( ps /\ ph ) -> ps )
4	3	a1i 9	. . 3 |- ( ph -> ( ( ps /\ ph ) -> ps ) )
5		pm2.24 622	. . . 4 |- ( ph -> ( -. ph -> ps ) )
6	5	adantld 278	. . 3 |- ( ph -> ( ( ch /\ -. ph ) -> ps ) )
7	4, 6	jaod 718	. 2 |- ( ph -> ( ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) -> ps ) )
8	2, 7	impbid 129	1 |- ( ph -> ( ps <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  iftrue  3566