Theorem bj-ceqsaltv 32876

Description: Version of bj-ceqsalt 32875 with a dv condition on x , V, removing dependency on df-sb 1881 and df-clab 2609. Prefer its use over bj-ceqsalt 32875 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

Step	Hyp	Ref	Expression
1		bj-elissetv 32861	. . 3 |- ( A e. V -> E. x x = A )
2	1	3anim3i 1250	. 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 32873	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ E. x x = A ) -> ( A. x ( x = A -> ph ) <-> ps ) )
4	2, 3	syl 17	1 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-ex 1705  df-nf 1710  df-clel 2618

This theorem is referenced by:  bj-ceqsalgvALT  32881