Theorem dedlema 920

Description: Lemma for weak deduction theorem. (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 374	. . 3 |- ((ps /\ ph) -> ((ps /\ ph) \/ (ch /\ -. ph)))
2	1	expcom 424	. 2 |- (ph -> (ps -> ((ps /\ ph) \/ (ch /\ -. ph))))
3		simpl 443	. . . 4 |- ((ps /\ ph) -> ps)
4	3	a1i 10	. . 3 |- (ph -> ((ps /\ ph) -> ps))
5		pm2.24 101	. . . 4 |- (ph -> (-. ph -> ps))
6	5	adantld 453	. . 3 |- (ph -> ((ch /\ -. ph) -> ps))
7	4, 6	jaod 369	. 2 |- (ph -> (((ps /\ ph) \/ (ch /\ -. ph)) -> ps))
8	2, 7	impbid 183	1 |- (ph -> (ps <-> ((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  dedt  923  pm4.42  926