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) Avoid df-clab . (Revised by Wolf Lammen, 23-Jan-2025)

	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	19.23	$$⊢\forall x\left(x=A\to \psi \right)\leftrightarrow \left(\exists xx=A\to \psi \right)$$
5	3	pm5.74i	$$⊢\left(x=A\to \phi \right)\leftrightarrow \left(x=A\to \psi \right)$$
6	5	albii	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \forall x\left(x=A\to \psi \right)$$
7	2	isseti	$$⊢\exists xx=A$$
8	7	a1bi	$$⊢\psi \leftrightarrow \left(\exists xx=A\to \psi \right)$$
9	4 6 8	3bitr4i	$$⊢\forall x\left(x=A\to \phi \right)\leftrightarrow \psi$$