Theorem dedlem0b 1001

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		pm2.21 120	. . . 4 |- ( -. ph -> ( ph -> ( ch /\ ph ) ) )
2	1	imim2d 57	. . 3 |- ( -. ph -> ( ( ps -> ph ) -> ( ps -> ( ch /\ ph ) ) ) )
3	2	com23 86	. 2 |- ( -. ph -> ( ps -> ( ( ps -> ph ) -> ( ch /\ ph ) ) ) )
4		pm2.21 120	. . . . 5 |- ( -. ps -> ( ps -> ph ) )
5		simpr 477	. . . . 5 |- ( ( ch /\ ph ) -> ph )
6	4, 5	imim12i 62	. . . 4 |- ( ( ( ps -> ph ) -> ( ch /\ ph ) ) -> ( -. ps -> ph ) )
7	6	con1d 139	. . 3 |- ( ( ( ps -> ph ) -> ( ch /\ ph ) ) -> ( -. ph -> ps ) )
8	7	com12 32	. 2 |- ( -. ph -> ( ( ( ps -> ph ) -> ( ch /\ ph ) ) -> ps ) )
9	3, 8	impbid 202	1 |- ( -. ph -> ( ps <-> ( ( ps -> ph ) -> ( ch /\ ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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-an 386

This theorem is referenced by: (None)