Theorem bj-stim 13781

Description: A conjunction with a stable consequent is stable. See stabnot 828 for negation , bj-stan 13782 for conjunction , and bj-stal 13784 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-stim	|- (STAB ps -> STAB ( ph -> ps ) )

Proof of Theorem bj-stim

Step	Hyp	Ref	Expression
1		bj-nnim 13770	. . 3 |- ( -. -. ( ph -> ps ) -> ( ph -> -. -. ps ) )
2		imim2 55	. . 3 |- ( ( -. -. ps -> ps ) -> ( ( ph -> -. -. ps ) -> ( ph -> ps ) ) )
3	1, 2	syl5 32	. 2 |- ( ( -. -. ps -> ps ) -> ( -. -. ( ph -> ps ) -> ( ph -> ps ) ) )
4		df-stab 826	. 2 |- (STAB ps <-> ( -. -. ps -> ps ) )
5		df-stab 826	. 2 |- (STAB ( ph -> ps ) <-> ( -. -. ( ph -> ps ) -> ( ph -> ps ) ) )
6	3, 4, 5	3imtr4i 200	1 |- (STAB ps -> STAB ( ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-stab 826

This theorem is referenced by: (None)