Theorem dedlema 1002

Description: Lemma for weak deduction theorem. See also ifptru 1023. (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 400	. . 3 |- ( ( ps /\ ph ) -> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) )
2	1	expcom 451	. 2 |- ( ph -> ( ps -> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )
3		simpl 473	. . . 4 |- ( ( ps /\ ph ) -> ps )
4	3	a1i 11	. . 3 |- ( ph -> ( ( ps /\ ph ) -> ps ) )
5		pm2.24 121	. . . 4 |- ( ph -> ( -. ph -> ps ) )
6	5	adantld 483	. . 3 |- ( ph -> ( ( ch /\ -. ph ) -> ps ) )
7	4, 6	jaod 395	. 2 |- ( ph -> ( ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) -> ps ) )
8	2, 7	impbid 202	1 |- ( ph -> ( ps <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  pm4.42  1004  elimhOLD  1033  dedtOLD  1034