Theorem bj-ceqsalg

Description: Remove from ceqsalg dependency on ax-ext (and on df-cleq and df-v ). See also bj-ceqsalgv . (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-ceqsalg.1	$$⊢Ⅎx\psi$$
2		bj-ceqsalg.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		bj-elisset	$$⊢A\in V\to \exists xx=A$$
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	$$⊢A\in V\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$