Theorem bj-stan 13628

Description: The conjunction of two stable formulas is stable. See bj-stim 13627 for implication, stabnot 823 for negation, and bj-stal 13630 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-stan	|- ( (STAB ph /\ STAB ps ) -> STAB ( ph /\ ps ) )

Proof of Theorem bj-stan

Step	Hyp	Ref	Expression
1		bj-nnan 13617	. . 3 |- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )
2		anim12 342	. . 3 |- ( ( ( -. -. ph -> ph ) /\ ( -. -. ps -> ps ) ) -> ( ( -. -. ph /\ -. -. ps ) -> ( ph /\ ps ) ) )
3	1, 2	syl5 32	. 2 |- ( ( ( -. -. ph -> ph ) /\ ( -. -. ps -> ps ) ) -> ( -. -. ( ph /\ ps ) -> ( ph /\ ps ) ) )
4		df-stab 821	. . 3 |- (STAB ph <-> ( -. -. ph -> ph ) )
5		df-stab 821	. . 3 |- (STAB ps <-> ( -. -. ps -> ps ) )
6	4, 5	anbi12i 456	. 2 |- ( (STAB ph /\ STAB ps ) <-> ( ( -. -. ph -> ph ) /\ ( -. -. ps -> ps ) ) )
7		df-stab 821	. 2 |- (STAB ( ph /\ ps ) <-> ( -. -. ( ph /\ ps ) -> ( ph /\ ps ) ) )
8	3, 6, 7	3imtr4i 200	1 |- ( (STAB ph /\ STAB ps ) -> STAB ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103  STAB wstab 820

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 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-stab 821

This theorem is referenced by: (None)