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Theorem bj-stim 12939
Description: A conjunction with a stable consequent is stable. See stabnot 818 for negation and bj-stan 12940 for conjunction. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-stim  |-  (STAB  ps  -> STAB  ( ph  ->  ps ) )

Proof of Theorem bj-stim
StepHypRef Expression
1 bj-nnim 12932 . . 3  |-  ( -. 
-.  ( ph  ->  ps )  ->  ( ph  ->  -.  -.  ps )
)
2 imim2 55 . . 3  |-  ( ( -.  -.  ps  ->  ps )  ->  ( ( ph  ->  -.  -.  ps )  ->  ( ph  ->  ps ) ) )
31, 2syl5 32 . 2  |-  ( ( -.  -.  ps  ->  ps )  ->  ( -.  -.  ( ph  ->  ps )  ->  ( ph  ->  ps ) ) )
4 df-stab 816 . 2  |-  (STAB  ps  <->  ( -.  -.  ps  ->  ps )
)
5 df-stab 816 . 2  |-  (STAB  ( ph  ->  ps )  <->  ( -.  -.  ( ph  ->  ps )  ->  ( ph  ->  ps ) ) )
63, 4, 53imtr4i 200 1  |-  (STAB  ps  -> STAB  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  STAB wstab 815
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
This theorem depends on definitions:  df-bi 116  df-stab 816
This theorem is referenced by: (None)
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