Theorem dvelimh 2336

Description: Version of dvelim 2337 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Hypotheses

Ref	Expression
dvelimh.1	|- ( ph -> A. x ph )
dvelimh.2	|- ( ps -> A. z ps )
dvelimh.3	|- ( z = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
dvelimh	|- ( -. A. x x = y -> ( ps -> A. x ps ) )

Proof of Theorem dvelimh

Step	Hyp	Ref	Expression
1		dvelimh.1	. . . 4 |- ( ph -> A. x ph )
2	1	nf5i 2024	. . 3 |- F/ x ph
3		dvelimh.2	. . . 4 |- ( ps -> A. z ps )
4	3	nf5i 2024	. . 3 |- F/ z ps
5		dvelimh.3	. . 3 |- ( z = y -> ( ph <-> ps ) )
6	2, 4, 5	dvelimf 2334	. 2 |- ( -. A. x x = y -> F/ x ps )
7	6	nf5rd 2066	1 |- ( -. A. x x = y -> ( ps -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481

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:  dvelim  2337  dveeq1-o16  34221  dveel2ALT  34224  ax6e2nd  38774  ax6e2ndVD  39144  ax6e2ndALT  39166