Theorem rexralbidv

Description: Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006)

	Ref	Expression
Hypothesis	2rexbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rexralbidv	$$⊢\phi \to \left(\exists x\in A\forall y\in B\psi \leftrightarrow \exists x\in A\forall y\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		2rexbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	ralbidv	$$⊢\phi \to \left(\forall y\in B\psi \leftrightarrow \forall y\in B\chi \right)$$
3	2	rexbidv	$$⊢\phi \to \left(\exists x\in A\forall y\in B\psi \leftrightarrow \exists x\in A\forall y\in B\chi \right)$$