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	$$⊢B\in V$$
Assertion	bj-rexcom4bv	$$⊢\exists x\exists y\in A\left(\phi \wedge x=B\right)\leftrightarrow \exists y\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-rexcom4bv.1	$$⊢B\in V$$
2		rexcom4a	$$⊢\exists x\exists y\in A\left(\phi \wedge x=B\right)\leftrightarrow \exists y\in A\left(\phi \wedge \exists xx=B\right)$$
3	1	bj-issetiv	$$⊢\exists xx=B$$
4	3	biantru	$$⊢\phi \leftrightarrow \left(\phi \wedge \exists xx=B\right)$$
5	4	rexbii	$$⊢\exists y\in A\phi \leftrightarrow \exists y\in A\left(\phi \wedge \exists xx=B\right)$$
6	2 5	bitr4i	$$⊢\exists x\exists y\in A\left(\phi \wedge x=B\right)\leftrightarrow \exists y\in A\phi$$