Theorem const 837

Description: Contraposition of a stable proposition. See comment of condc 838. (Contributed by BJ, 18-Nov-2023.)

Assertion

Ref	Expression
const	|- (STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )

Proof of Theorem const

Step	Hyp	Ref	Expression
1		df-stab 816	. 2 |- (STAB ph <-> ( -. -. ph -> ph ) )
2		con3 631	. . 3 |- ( ( -. ph -> -. ps ) -> ( -. -. ps -> -. -. ph ) )
3		notnot 618	. . . 4 |- ( ps -> -. -. ps )
4		imim2 55	. . . 4 |- ( ( -. -. ph -> ph ) -> ( ( -. -. ps -> -. -. ph ) -> ( -. -. ps -> ph ) ) )
5	3, 4	syl7 69	. . 3 |- ( ( -. -. ph -> ph ) -> ( ( -. -. ps -> -. -. ph ) -> ( ps -> ph ) ) )
6	2, 5	syl5 32	. 2 |- ( ( -. -. ph -> ph ) -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
7	1, 6	sylbi 120	1 |- (STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-stab 816

This theorem is referenced by:  condc  838