Theorem bj-ceqsal

Description: Remove from ceqsal dependency on ax-ext (and on df-cleq , df-v , df-clab , df-sb ). (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-ceqsal.1	$$⊢Ⅎx\psi$$
	bj-ceqsal.2	$$⊢A\in \mathrm{V}$$
	bj-ceqsal.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	bj-ceqsal	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-ceqsal.1	$$⊢Ⅎx\psi$$
2		bj-ceqsal.2	$$⊢A\in \mathrm{V}$$
3		bj-ceqsal.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
4	1 3	bj-ceqsalgv	$$⊢A\in \mathrm{V}\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$
5	2 4	ax-mp	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \psi$$