Theorem ceqsralv

Description: Restricted quantifier version of ceqsalv . (Contributed by NM, 21-Jun-2013)

	Ref	Expression
Hypothesis	ceqsralv.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	ceqsralv	$$⊢A\in B\to \left(\forall x\in B\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ceqsralv.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎx\psi$$
3	1	ax-gen	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)$$
4		ceqsralt	$$⊢\left(Ⅎx\psi \wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\wedge A\in B\right)\to \left(\forall x\in B\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$
5	2 3 4	mp3an12	$$⊢A\in B\to \left(\forall x\in B\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$