Theorem bj-stan 14639

Description: The conjunction of two stable formulas is stable. See bj-stim 14638 for implication, stabnot 833 for negation, and bj-stal 14641 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 14628	. . 3 |- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )
2		anim12 344	. . 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 831	. . 3 |- (STAB ph <-> ( -. -. ph -> ph ) )
5		df-stab 831	. . 3 |- (STAB ps <-> ( -. -. ps -> ps ) )
6	4, 5	anbi12i 460	. 2 |- ( (STAB ph /\ STAB ps ) <-> ( ( -. -. ph -> ph ) /\ ( -. -. ps -> ps ) ) )
7		df-stab 831	. 2 |- (STAB ( ph /\ ps ) <-> ( -. -. ( ph /\ ps ) -> ( ph /\ ps ) ) )
8	3, 6, 7	3imtr4i 201	1 |- ( (STAB ph /\ STAB ps ) -> STAB ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  STAB wstab 830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-stab 831

This theorem is referenced by: (None)