Theorem bj-stal 13128

Description: The universal quantification of stable formula is stable. See bj-stim 13125 for implication, stabnot 819 for negation, and bj-stan 13126 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		bj-nnal 13120	. . 3 |- ( -. -. A. x ph -> A. x -. -. ph )
2		alim 1434	. . 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 817	. . 3 |- (STAB ph <-> ( -. -. ph -> ph ) )
5	4	albii 1447	. 2 |- ( A. xSTAB ph <-> A. x ( -. -. ph -> ph ) )
6		df-stab 817	. 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 816   A.wal 1330

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  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-stab 817  df-tru 1335  df-fal 1338  df-nf 1438

This theorem is referenced by: (None)