Theorem con3OLD 1035

Description: Old version of con3ALT 1032. Obsolete as of 16-Mar-2021. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
con3OLD	|- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )

Proof of Theorem con3OLD

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( ( ps <-> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) -> ( ps <-> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) )
2	1	notbid 308	. . 3 |- ( ( ps <-> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) -> ( -. ps <-> -. ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) )
3	2	imbi1d 331	. 2 |- ( ( ps <-> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) -> ( ( -. ps -> -. ph ) <-> ( -. ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) -> -. ph ) ) )
4	1	imbi2d 330	. . . 4 |- ( ( ps <-> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) -> ( ( ph -> ps ) <-> ( ph -> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) ) )
5		id 22	. . . . 5 |- ( ( ph <-> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) -> ( ph <-> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) )
6	5	imbi2d 330	. . . 4 |- ( ( ph <-> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) -> ( ( ph -> ph ) <-> ( ph -> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) ) ) )
7		id 22	. . . 4 |- ( ph -> ph )
8	4, 6, 7	elimhOLD 1033	. . 3 |- ( ph -> ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) )
9	8	con3i 150	. 2 |- ( -. ( ( ps /\ ( ph -> ps ) ) \/ ( ph /\ -. ( ph -> ps ) ) ) -> -. ph )
10	3, 9	dedtOLD 1034	1 |- ( ( ph -> ps ) -> ( -. ps -> -. 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: (None)