Theorem dedlemb 921

Description: Lemma for weak deduction theorem. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Assertion

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

Proof of Theorem dedlemb

Step	Hyp	Ref	Expression
1		olc 373	. . 3 |- ((ch /\ -. ph) -> ((ps /\ ph) \/ (ch /\ -. ph)))
2	1	expcom 424	. 2 |- (-. ph -> (ch -> ((ps /\ ph) \/ (ch /\ -. ph))))
3		pm2.21 100	. . . 4 |- (-. ph -> (ph -> ch))
4	3	adantld 453	. . 3 |- (-. ph -> ((ps /\ ph) -> ch))
5		simpl 443	. . . 4 |- ((ch /\ -. ph) -> ch)
6	5	a1i 10	. . 3 |- (-. ph -> ((ch /\ -. ph) -> ch))
7	4, 6	jaod 369	. 2 |- (-. ph -> (((ps /\ ph) \/ (ch /\ -. ph)) -> ch))
8	2, 7	impbid 183	1 |- (-. ph -> (ch <-> ((ps /\ ph) \/ (ch /\ -. ph))))

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:  elimh  922  pm4.42  926  iffalse  3670