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Theorem bj-ceqsaltv 31497
Description: Version of bj-ceqsalt 31496 with a dv condition on  x ,  V, removing dependency on df-sb 1800 and df-clab 2440. Prefer its use over bj-ceqsalt 31496 when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ceqsaltv  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable groups:    x, A    x, V
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem bj-ceqsaltv
StepHypRef Expression
1 bj-elissetv 31482 . . 3  |-  ( A  e.  V  ->  E. x  x  =  A )
213anim3i 1197 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  E. x  x  =  A
) )
3 bj-ceqsalt0 31494 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  E. x  x  =  A
)  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
42, 3syl 17 1  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ w3a 986   A.wal 1444    = wceq 1446   E.wex 1665   F/wnf 1669    e. wcel 1889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-12 1935
This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 988  df-ex 1666  df-nf 1670  df-clel 2449
This theorem is referenced by:  bj-ceqsalgvALT  31502
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