Theorem bj-ceqsalgv

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

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

Proof

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