Theorem bj-ceqsalgv

Description: Version of bj-ceqsalg with a disjoint variable condition on x , V , removing dependency on df-sb and df-clab . Prefer its use over bj-ceqsalg when sufficient (in particular when V is substituted for _V ). (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-ceqsalgv.1	$$⊢Ⅎx\psi$$
	bj-ceqsalgv.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	bj-ceqsalgv	$$⊢A\in V\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-ceqsalgv.1	$$⊢Ⅎx\psi$$
2		bj-ceqsalgv.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		elissetv	$$⊢A\in V\to \exists xx=A$$
4	1 2	bj-ceqsalg0	$$⊢\exists xx=A\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$
5	3 4	syl	$$⊢A\in V\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$