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Theorem dvelimf 1933
Description: Version of dvelim 1935 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
Hypotheses
Ref Expression
dvelimf.1  |-  ( ph  ->  A. x ph )
dvelimf.2  |-  ( ps 
->  A. z ps )
dvelimf.3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
dvelimf  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)

Proof of Theorem dvelimf
StepHypRef Expression
1 dvelimf.1 . . 3  |-  ( ph  ->  A. x ph )
21hbsb4 1930 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  z ]
ph  ->  A. x [ y  /  z ] ph ) )
3 dvelimf.2 . . 3  |-  ( ps 
->  A. z ps )
4 dvelimf.3 . . 3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
53, 4sbieh 1714 . 2  |-  ( [ y  /  z ]
ph 
<->  ps )
65albii 1400 . 2  |-  ( A. x [ y  /  z ] ph  <->  A. x ps )
72, 5, 63imtr3g 202 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103   A.wal 1283   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  dvelim  1935  dveel1  1938  dveel2  1939
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