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Theorem bj-stal 12946
Description: The universal quantification of stable formula is stable. See bj-stim 12943 for implication, stabnot 818 for negation, and bj-stan 12944 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
StepHypRef Expression
1 bj-nnal 12938 . . 3  |-  ( -. 
-.  A. x ph  ->  A. x  -.  -.  ph )
2 alim 1433 . . 3  |-  ( A. x ( -.  -.  ph 
->  ph )  ->  ( A. x  -.  -.  ph  ->  A. x ph )
)
31, 2syl5 32 . 2  |-  ( A. x ( -.  -.  ph 
->  ph )  ->  ( -.  -.  A. x ph  ->  A. x ph )
)
4 df-stab 816 . . 3  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
54albii 1446 . 2  |-  ( A. xSTAB  ph  <->  A. x ( -.  -.  ph 
->  ph ) )
6 df-stab 816 . 2  |-  (STAB  A. x ph 
<->  ( -.  -.  A. x ph  ->  A. x ph ) )
73, 5, 63imtr4i 200 1  |-  ( A. xSTAB  ph 
-> STAB  A. x ph )
Colors of variables: wff set class
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)
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