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Theorem bj-stan 12940
Description: The conjunction of two stable formulas is stable. See bj-stim 12939 for implication, stabnot 818 for negation, and bj-stal 12942 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-stan  |-  ( (STAB  ph  /\ STAB  ps )  -> STAB  ( ph  /\  ps ) )

Proof of Theorem bj-stan
StepHypRef Expression
1 bj-nnan 12933 . . 3  |-  ( -. 
-.  ( ph  /\  ps )  ->  ( -. 
-.  ph  /\  -.  -.  ps ) )
2 anim12 341 . . 3  |-  ( ( ( -.  -.  ph  ->  ph )  /\  ( -.  -.  ps  ->  ps ) )  ->  (
( -.  -.  ph  /\ 
-.  -.  ps )  ->  ( ph  /\  ps ) ) )
31, 2syl5 32 . 2  |-  ( ( ( -.  -.  ph  ->  ph )  /\  ( -.  -.  ps  ->  ps ) )  ->  ( -.  -.  ( ph  /\  ps )  ->  ( ph  /\ 
ps ) ) )
4 df-stab 816 . . 3  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
5 df-stab 816 . . 3  |-  (STAB  ps  <->  ( -.  -.  ps  ->  ps )
)
64, 5anbi12i 455 . 2  |-  ( (STAB  ph  /\ STAB  ps ) 
<->  ( ( -.  -.  ph 
->  ph )  /\  ( -.  -.  ps  ->  ps ) ) )
7 df-stab 816 . 2  |-  (STAB  ( ph  /\ 
ps )  <->  ( -.  -.  ( ph  /\  ps )  ->  ( ph  /\  ps ) ) )
83, 6, 73imtr4i 200 1  |-  ( (STAB  ph  /\ STAB  ps )  -> STAB  ( ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103  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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