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Theorem bj-ceqsaltv 34638
Description: Version of bj-ceqsalt 34637 with a dv condition on  x ,  V, removing dependency on df-sb 1741 and df-clab 2443. Prefer its use over bj-ceqsalt 34637 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 34623 . . 3  |-  ( A  e.  V  ->  E. x  x  =  A )
213anim3i 1184 . 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 34635 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  E. x  x  =  A
)  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
42, 3syl 16 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 184    /\ w3a 973   A.wal 1393    = wceq 1395   E.wex 1613   F/wnf 1617    e. wcel 1819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1614  df-nf 1618  df-clel 2452
This theorem is referenced by:  bj-ceqsalgvALT  34643
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