Theorem ceqsal

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

Proof

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