Theorem bj-stal 13630

Description: The universal quantification of a stable formula is stable. See bj-stim 13627 for implication, stabnot 823 for negation, and bj-stan 13628 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 1637	. . 3 |- ( -. -. A. x ph -> A. x -. -. ph )
2		alim 1445	. . 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 821	. . 3 |- (STAB ph <-> ( -. -. ph -> ph ) )
5	4	albii 1458	. 2 |- ( A. xSTAB ph <-> A. x ( -. -. ph -> ph ) )
6		df-stab 821	. 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 820   A.wal 1341

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-stab 821  df-tru 1346  df-fal 1349  df-nf 1449

This theorem is referenced by: (None)