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 -> ( ph <-> ps ) )
Assertion	ralsng	|- ( A e. V -> ( A. x e. { A } ph <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		ralsng.1	|- ( x = A -> ( ph <-> ps ) )
2		nfv	|- F/ x ps
3	2 1	ralsngf	|- ( A e. V -> ( A. x e. { A } ph <-> ps ) )