Theorem bj-ceqsalgALT

Description: Alternate proof of bj-ceqsalg . (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

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