Theorem dedlem0b 919

Description: Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)

Assertion

Ref	Expression
dedlem0b	|- (-. ph -> (ps <-> ((ps -> ph) -> (ch /\ ph))))

Proof of Theorem dedlem0b

Step	Hyp	Ref	Expression
1		pm2.21 100	. . . 4 |- (-. ph -> (ph -> (ch /\ ph)))
2	1	imim2d 48	. . 3 |- (-. ph -> ((ps -> ph) -> (ps -> (ch /\ ph))))
3	2	com23 72	. 2 |- (-. ph -> (ps -> ((ps -> ph) -> (ch /\ ph))))
4		pm2.21 100	. . . . 5 |- (-. ps -> (ps -> ph))
5		simpr 447	. . . . 5 |- ((ch /\ ph) -> ph)
6	4, 5	imim12i 53	. . . 4 |- (((ps -> ph) -> (ch /\ ph)) -> (-. ps -> ph))
7	6	con1d 116	. . 3 |- (((ps -> ph) -> (ch /\ ph)) -> (-. ph -> ps))
8	7	com12 27	. 2 |- (-. ph -> (((ps -> ph) -> (ch /\ ph)) -> ps))
9	3, 8	impbid 183	1 |- (-. ph -> (ps <-> ((ps -> ph) -> (ch /\ ph))))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ 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-an 360

This theorem is referenced by: (None)