Theorem ceqsalg

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT . (Contributed by NM, 29-Oct-2003) (Proof shortened by BJ, 29-Sep-2019)

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

Proof

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