Theorem bj-stal 13784

Description: The universal quantification of a stable formula is stable. See bj-stim 13781 for implication, stabnot 828 for negation, and bj-stan 13782 for conjunction. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-stal	|- ( A. xSTAB ph -> STAB A. x ph )

Proof of Theorem bj-stal

Step	Hyp	Ref	Expression
1		nnal 1642	. . 3 |- ( -. -. A. x ph -> A. x -. -. ph )
2		alim 1450	. . 3 |- ( A. x ( -. -. ph -> ph ) -> ( A. x -. -. ph -> A. x ph ) )
3	1, 2	syl5 32	. 2 |- ( A. x ( -. -. ph -> ph ) -> ( -. -. A. x ph -> A. x ph ) )
4		df-stab 826	. . 3 |- (STAB ph <-> ( -. -. ph -> ph ) )
5	4	albii 1463	. 2 |- ( A. xSTAB ph <-> A. x ( -. -. ph -> ph ) )
6		df-stab 826	. 2 |- (STAB A. x ph <-> ( -. -. A. x ph -> A. x ph ) )
7	3, 5, 6	3imtr4i 200	1 |- ( A. xSTAB ph -> STAB A. x ph )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 825   A.wal 1346

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  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-stab 826  df-tru 1351  df-fal 1354  df-nf 1454

This theorem is referenced by: (None)