Theorem bj-ceqsalgv 32880

Description: Version of bj-ceqsalg 32878 with a dv condition on x , V, removing dependency on df-sb 1881 and df-clab 2609. Prefer its use over bj-ceqsalg 32878 when sufficient (in particular when V is substituted for _V). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-ceqsalgv.1	|- F/ x ps
bj-ceqsalgv.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
bj-ceqsalgv	|- ( 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-ceqsalgv

Step	Hyp	Ref	Expression
1		bj-elissetv 32861	. 2 |- ( A e. V -> E. x x = A )
2		bj-ceqsalgv.1	. . 3 |- F/ x ps
3		bj-ceqsalgv.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	2, 3	bj-ceqsalg0 32877	. 2 |- ( E. x x = A -> ( A. x ( x = A -> ph ) <-> ps ) )
5	1, 4	syl 17	1 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   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-ceqsal  32882  bj-raldifsn  33054