Theorem dvelimf

Description: Version of dvelimv without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 1-Oct-2002) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvelimf.1	|- F/ x ph
	dvelimf.2	|- F/ z ps
	dvelimf.3	|- ( z = y -> ( ph <-> ps ) )
Assertion	dvelimf	|- ( -. A. x x = y -> F/ x ps )

Proof

Step	Hyp	Ref	Expression
1		dvelimf.1	|- F/ x ph
2		dvelimf.2	|- F/ z ps
3		dvelimf.3	|- ( z = y -> ( ph <-> ps ) )
4	2 3	equsal	|- ( A. z ( z = y -> ph ) <-> ps )
5	4	bicomi	|- ( ps <-> A. z ( z = y -> ph ) )
6		nfnae	|- F/ z -. A. x x = y
7		nfeqf	|- ( ( -. A. x x = z /\ -. A. x x = y ) -> F/ x z = y )
8	7	ancoms	|- ( ( -. A. x x = y /\ -. A. x x = z ) -> F/ x z = y )
9	1	a1i	|- ( ( -. A. x x = y /\ -. A. x x = z ) -> F/ x ph )
10	8 9	nfimd	|- ( ( -. A. x x = y /\ -. A. x x = z ) -> F/ x ( z = y -> ph ) )
11	6 10	nfald2	|- ( -. A. x x = y -> F/ x A. z ( z = y -> ph ) )
12	5 11	nfxfrd	|- ( -. A. x x = y -> F/ x ps )