Theorem dedlema 910

Description: Lemma for iftrue 3356. (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 665	. . 3 |- ( ( ps /\ ph ) -> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) )
2	1	expcom 114	. 2 |- ( ph -> ( ps -> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )
3		simpl 107	. . . 4 |- ( ( ps /\ ph ) -> ps )
4	3	a1i 9	. . 3 |- ( ph -> ( ( ps /\ ph ) -> ps ) )
5		pm2.24 583	. . . 4 |- ( ph -> ( -. ph -> ps ) )
6	5	adantld 272	. . 3 |- ( ph -> ( ( ch /\ -. ph ) -> ps ) )
7	4, 6	jaod 669	. 2 |- ( ph -> ( ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) -> ps ) )
8	2, 7	impbid 127	1 |- ( ph -> ( ps <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  iftrue  3356