Theorem bj-stim 16786

Description: A conjunction with a stable consequent is stable. See stabnot 845 for negation , bj-stan 16787 for conjunction , and bj-stal 16789 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 16775	. . 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 843	. 2 |- (STAB ps <-> ( -. -. ps -> ps ) )
5		df-stab 843	. 2 |- (STAB ( ph -> ps ) <-> ( -. -. ( ph -> ps ) -> ( ph -> ps ) ) )
6	3, 4, 5	3imtr4i 201	1 |- (STAB ps -> STAB ( ph -> ps ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4  STAB wstab 842

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-stab 843

This theorem is used by: (None)