Theorem dvelimf 2334

Description: Version of dvelimv 2338 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Hypotheses

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

Assertion

Ref	Expression
dvelimf	|- ( -. A. x x = y -> F/ x ps )

Proof of Theorem dvelimf

Step	Hyp	Ref	Expression
1		dvelimf.2	. . . 4 |- F/ z ps
2		dvelimf.3	. . . 4 |- ( z = y -> ( ph <-> ps ) )
3	1, 2	equsal 2291	. . 3 |- ( A. z ( z = y -> ph ) <-> ps )
4	3	bicomi 214	. 2 |- ( ps <-> A. z ( z = y -> ph ) )
5		nfnae 2318	. . 3 |- F/ z -. A. x x = y
6		nfeqf 2301	. . . . 5 |- ( ( -. A. x x = z /\ -. A. x x = y ) -> F/ x z = y )
7	6	ancoms 469	. . . 4 |- ( ( -. A. x x = y /\ -. A. x x = z ) -> F/ x z = y )
8		dvelimf.1	. . . . 5 |- F/ x ph
9	8	a1i 11	. . . 4 |- ( ( -. A. x x = y /\ -. A. x x = z ) -> F/ x ph )
10	7, 9	nfimd 1823	. . 3 |- ( ( -. A. x x = y /\ -. A. x x = z ) -> F/ x ( z = y -> ph ) )
11	5, 10	nfald2 2331	. 2 |- ( -. A. x x = y -> F/ x A. z ( z = y -> ph ) )
12	4, 11	nfxfrd 1780	1 |- ( -. A. x x = y -> F/ x ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  dvelimdf  2335  dvelimh  2336  dvelimnf  2339