Theorem ralsnsg

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

	Ref	Expression
Assertion	ralsnsg	|- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Proof

Step	Hyp	Ref	Expression
1		sbc6g	|- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
2		df-ral	|- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
3		velsn	|- ( x e. { A } <-> x = A )
4	3	imbi1i	|- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
5	4	albii	|- ( A. x ( x e. { A } -> ph ) <-> A. x ( x = A -> ph ) )
6	2 5	bitri	|- ( A. x e. { A } ph <-> A. x ( x = A -> ph ) )
7	1 6	syl6rbbr	|- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )