Metamath Proof Explorer


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 )