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Theorem bj-stand 12945
Description: The conjunction of two stable formulas is stable. Deduction form of bj-stan 12944. Its proof is shorter, so one could prove it first and then bj-stan 12944 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-stand.1  |-  ( ph  -> STAB  ps )
bj-stand.2  |-  ( ph  -> STAB  ch )
Assertion
Ref Expression
bj-stand  |-  ( ph  -> STAB  ( ps  /\  ch )
)

Proof of Theorem bj-stand
StepHypRef Expression
1 bj-nnan 12937 . . 3  |-  ( -. 
-.  ( ps  /\  ch )  ->  ( -. 
-.  ps  /\  -.  -.  ch ) )
2 bj-stand.1 . . . . 5  |-  ( ph  -> STAB  ps )
3 df-stab 816 . . . . 5  |-  (STAB  ps  <->  ( -.  -.  ps  ->  ps )
)
42, 3sylib 121 . . . 4  |-  ( ph  ->  ( -.  -.  ps  ->  ps ) )
5 bj-stand.2 . . . . 5  |-  ( ph  -> STAB  ch )
6 df-stab 816 . . . . 5  |-  (STAB  ch  <->  ( -.  -.  ch  ->  ch )
)
75, 6sylib 121 . . . 4  |-  ( ph  ->  ( -.  -.  ch  ->  ch ) )
84, 7anim12d 333 . . 3  |-  ( ph  ->  ( ( -.  -.  ps  /\  -.  -.  ch )  ->  ( ps  /\  ch ) ) )
91, 8syl5 32 . 2  |-  ( ph  ->  ( -.  -.  ( ps  /\  ch )  -> 
( ps  /\  ch ) ) )
10 df-stab 816 . 2  |-  (STAB  ( ps 
/\  ch )  <->  ( -.  -.  ( ps  /\  ch )  ->  ( ps  /\  ch ) ) )
119, 10sylibr 133 1  |-  ( ph  -> STAB  ( ps  /\  ch )
)
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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