Theorem bj-ceqsalg0

Description: The FOL content of ceqsalg . (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-ceqsalg0.1	$$⊢Ⅎx\psi$$
	bj-ceqsalg0.2	$$⊢\chi \to \left(\phi \leftrightarrow \psi \right)$$
Assertion	bj-ceqsalg0	$$⊢\exists x\chi \to \left(\forall x\left(\chi \to \phi \right)\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-ceqsalg0.1	$$⊢Ⅎx\psi$$
2		bj-ceqsalg0.2	$$⊢\chi \to \left(\phi \leftrightarrow \psi \right)$$
3	2	ax-gen	$$⊢\forall x\left(\chi \to \left(\phi \leftrightarrow \psi \right)\right)$$
4		bj-ceqsalt0	$$⊢\left(Ⅎx\psi \wedge \forall x\left(\chi \to \left(\phi \leftrightarrow \psi \right)\right)\wedge \exists x\chi \right)\to \left(\forall x\left(\chi \to \phi \right)\leftrightarrow \psi \right)$$
5	1 3 4	mp3an12	$$⊢\exists x\chi \to \left(\forall x\left(\chi \to \phi \right)\leftrightarrow \psi \right)$$