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Theorem dvelimh 2185
Description: Version of dvelim 2186 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
StepHypRef Expression
1 dvelimh.1 . . . 4  |-  ( ph  ->  A. x ph )
21nfi 1682 . . 3  |-  F/ x ph
3 dvelimh.2 . . . 4  |-  ( ps 
->  A. z ps )
43nfi 1682 . . 3  |-  F/ z ps
5 dvelimh.3 . . 3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
62, 4, 5dvelimf 2183 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )
76nfrd 1973 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by:  dvelim  2186  dveeq1-o16  32571  dveel2ALT  32574  ax6e2nd  36995  ax6e2ndVD  37368  ax6e2ndALT  37390
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