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Theorem dvelimh 2072
Description: Version of dvelim 2074 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 1561 . . 3  |-  F/ x ph
3 dvelimh.2 . . . 4  |-  ( ps 
->  A. z ps )
43nfi 1561 . . 3  |-  F/ z ps
5 dvelimh.3 . . 3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
62, 4, 5dvelimf 2069 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )
76nfrd 1780 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178   A.wal 1550
This theorem is referenced by:  dvelim  2074  dveeq1-o16  2267  dveel2ALT  2270  a9e2nd  28647  a9e2ndVD  29022  a9e2ndALT  29044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555
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