Theorem ceqsalv

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993)

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

Proof

Step	Hyp	Ref	Expression
1		ceqsalv.1	$$⊢A\in \mathrm{V}$$
2		ceqsalv.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		nfv	$$⊢Ⅎx\psi$$
4	3 1 2	ceqsal	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \psi$$