Theorem bj-ceqsaltv

Description: Version of bj-ceqsalt with a disjoint variable condition on x , V , removing dependency on df-sb and df-clab . Prefer its use over bj-ceqsalt when sufficient (in particular when V is substituted for _V ). (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ceqsaltv	|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )

Proof

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