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) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 30-Sep-2024)

	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		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		elisset	|- ( A e. V -> E. x x = A )
8	1	pm5.74i	|- ( ( x = A -> ph ) <-> ( x = A -> ps ) )
9	8	albii	|- ( A. x ( x = A -> ph ) <-> A. x ( x = A -> ps ) )
10	9	a1i	|- ( E. x x = A -> ( A. x ( x = A -> ph ) <-> A. x ( x = A -> ps ) ) )
11		19.23v	|- ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) )
12	11	a1i	|- ( E. x x = A -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
13		pm5.5	|- ( E. x x = A -> ( ( E. x x = A -> ps ) <-> ps ) )
14	10 12 13	3bitrd	|- ( E. x x = A -> ( A. x ( x = A -> ph ) <-> ps ) )
15	7 14	syl	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )
16	6 15	bitrid	|- ( A e. V -> ( A. x e. { A } ph <-> ps ) )