Theorem con3ALT 1032

Description: Proof of con3 149 from its associated inference con3i 150 that illustrates the use of the weak deduction theorem dedt 1031. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
con3ALT	|- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )

Proof of Theorem con3ALT

Step	Hyp	Ref	Expression
1		bicom1 211	. . . 4 |- ( (if- ( ( ph -> ps ) , ps , ph ) <-> ps ) -> ( ps <-> if- ( ( ph -> ps ) , ps , ph ) ) )
2	1	notbid 308	. . 3 |- ( (if- ( ( ph -> ps ) , ps , ph ) <-> ps ) -> ( -. ps <-> -. if- ( ( ph -> ps ) , ps , ph ) ) )
3	2	imbi1d 331	. 2 |- ( (if- ( ( ph -> ps ) , ps , ph ) <-> ps ) -> ( ( -. ps -> -. ph ) <-> ( -. if- ( ( ph -> ps ) , ps , ph ) -> -. ph ) ) )
4	1	imbi2d 330	. . . 4 |- ( (if- ( ( ph -> ps ) , ps , ph ) <-> ps ) -> ( ( ph -> ps ) <-> ( ph -> if- ( ( ph -> ps ) , ps , ph ) ) ) )
5		bicom1 211	. . . . 5 |- ( (if- ( ( ph -> ps ) , ps , ph ) <-> ph ) -> ( ph <-> if- ( ( ph -> ps ) , ps , ph ) ) )
6	5	imbi2d 330	. . . 4 |- ( (if- ( ( ph -> ps ) , ps , ph ) <-> ph ) -> ( ( ph -> ph ) <-> ( ph -> if- ( ( ph -> ps ) , ps , ph ) ) ) )
7		id 22	. . . 4 |- ( ph -> ph )
8	4, 6, 7	elimh 1030	. . 3 |- ( ph -> if- ( ( ph -> ps ) , ps , ph ) )
9	8	con3i 150	. 2 |- ( -. if- ( ( ph -> ps ) , ps , ph ) -> -. ph )
10	3, 9	dedt 1031	1 |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196  if-wif 1012

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  df-ifp 1013

This theorem is referenced by: (None)