Theorem brralrspcev

Description: Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022)

	Ref	Expression
Assertion	brralrspcev	$$⊢\left(B\in X\wedge \forall y\in YARB\right)\to \exists x\in X\forall y\in YARx$$

Proof

Step	Hyp	Ref	Expression
1		breq2	$$⊢x=B\to \left(ARx\leftrightarrow ARB\right)$$
2	1	ralbidv	$$⊢x=B\to \left(\forall y\in YARx\leftrightarrow \forall y\in YARB\right)$$
3	2	rspcev	$$⊢\left(B\in X\wedge \forall y\in YARB\right)\to \exists x\in X\forall y\in YARx$$