Theorem dedlema 569

Description: Lemma for weak deduction theorem.

Assertion

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

Proof of Theorem dedlema

Step	Hyp	Ref	Expression
1		orc 225	. . . 4 |- (ps -> (ps \/ (ch /\ -. ph)))
2	1	a1i 7	. . 3 |- (ph -> (ps -> (ps \/ (ch /\ -. ph))))
3		idd 11	. . . 4 |- (ph -> (ps -> ps))
4		pm2.24 72	. . . . 5 |- (ph -> (-. ph -> ps))
5	4	adantld 307	. . . 4 |- (ph -> ((ch /\ -. ph) -> ps))
6	3, 5	jaod 329	. . 3 |- (ph -> ((ps \/ (ch /\ -. ph)) -> ps))
7	2, 6	impbid 397	. 2 |- (ph -> (ps <-> (ps \/ (ch /\ -. ph))))
8		iba 486	. . 3 |- (ph -> (ps <-> (ps /\ ph)))
9	8	orbi1d 467	. 2 |- (ph -> ((ps \/ (ch /\ -. ph)) <-> ((ps /\ ph) \/ (ch /\ -. ph))))
10	7, 9	bitrd 406	1 |- (ph -> (ps <-> ((ps /\ ph) \/ (ch /\ -. ph))))

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

This theorem is referenced by:  elimh 571  dedt 572  consensus 574  iftrue 1780

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

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