Theorem ralsng

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005) (Revised by Mario Carneiro, 23-Apr-2015) (Proof shortened by AV, 7-Apr-2023)

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

Proof

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