Theorem bj-rexcom4bv

Description: Version of rexcom4b and bj-rexcom4b with a disjoint variable condition on x , V , hence removing dependency on df-sb and df-clab (so that it depends on df-clel and df-rex only on top of first-order logic). Prefer its use over bj-rexcom4b when sufficient (in particular when V is substituted for _V ). Note the V in the hypothesis instead of _V . (Contributed by BJ, 14-Sep-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-rexcom4bv.1	|- B e. V
Assertion	bj-rexcom4bv	|- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )

Proof

Step	Hyp	Ref	Expression
1		bj-rexcom4bv.1	|- B e. V
2		rexcom4a	|- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ( ph /\ E. x x = B ) )
3	1	bj-issetiv	|- E. x x = B
4	3	biantru	|- ( ph <-> ( ph /\ E. x x = B ) )
5	4	rexbii	|- ( E. y e. A ph <-> E. y e. A ( ph /\ E. x x = B ) )
6	2 5	bitr4i	|- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )