Theorem dedlemb 570

Description: Lemma for weak deduction theorem.

Assertion

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

Proof of Theorem dedlemb

Step	Hyp	Ref	Expression
1		pm3.21 233	. . 3 |- (-. ph -> (ch -> (ch /\ -. ph)))
2		olc 224	. . 3 |- ((ch /\ -. ph) -> ((ps /\ ph) \/ (ch /\ -. ph)))
3	1, 2	syl6 23	. 2 |- (-. ph -> (ch -> ((ps /\ ph) \/ (ch /\ -. ph))))
4		pm2.21 71	. . . . 5 |- (-. ph -> (ph -> (ps -> ch)))
5	4	com23 32	. . . 4 |- (-. ph -> (ps -> (ph -> ch)))
6	5	imp3a 279	. . 3 |- (-. ph -> ((ps /\ ph) -> ch))
7		pm3.26 256	. . . 4 |- ((ch /\ -. ph) -> ch)
8	7	a1i 7	. . 3 |- (-. ph -> ((ch /\ -. ph) -> ch))
9	6, 8	jaod 329	. 2 |- (-. ph -> (((ps /\ ph) \/ (ch /\ -. ph)) -> ch))
10	3, 9	impbid 397	1 |- (-. ph -> (ch <-> ((ps /\ ph) \/ (ch /\ -. ph))))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196

This theorem is referenced by:  elimh 571  consensus 574  iffalse 1781

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

metamath.org