Theorem bj-stal 12957

Description: The universal quantification of stable formula is stable. See bj-stim 12954 for implication, stabnot 818 for negation, and bj-stan 12955 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 12949	. . 3 |- ( -. -. A. x ph -> A. x -. -. ph )
2		alim 1433	. . 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 816	. . 3 |- (STAB ph <-> ( -. -. ph -> ph ) )
5	4	albii 1446	. 2 |- ( A. xSTAB ph <-> A. x ( -. -. ph -> ph ) )
6		df-stab 816	. 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 815   A.wal 1329

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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-stab 816  df-tru 1334  df-fal 1337  df-nf 1437

This theorem is referenced by: (None)