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) Avoid ax-12 . (Revised by SN, 8-Sep-2024)

	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	2	pm5.74i	$$⊢\left(x=A\to \phi \right)\leftrightarrow \left(x=A\to \psi \right)$$
4	3	albii	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \forall x\left(x=A\to \psi \right)$$
5		19.23v	$$⊢\forall x\left(x=A\to \psi \right)\leftrightarrow \left(\exists xx=A\to \psi \right)$$
6	1	isseti	$$⊢\exists xx=A$$
7		pm5.5	$$⊢\exists xx=A\to \left(\left(\exists xx=A\to \psi \right)\leftrightarrow \psi \right)$$
8	6 7	ax-mp	$$⊢\left(\exists xx=A\to \psi \right)\leftrightarrow \psi$$
9	4 5 8	3bitri	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \psi$$