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Theorem dvelimdf 1940
Description: Deduction form of dvelimf 1939. This version may be useful if we want to avoid ax-17 1464 and use ax-16 1742 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
Hypotheses
Ref Expression
dvelimdf.1  |-  F/ x ph
dvelimdf.2  |-  F/ z
ph
dvelimdf.3  |-  ( ph  ->  F/ x ps )
dvelimdf.4  |-  ( ph  ->  F/ z ch )
dvelimdf.5  |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
dvelimdf  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )

Proof of Theorem dvelimdf
StepHypRef Expression
1 dvelimdf.2 . . . 4  |-  F/ z
ph
2 dvelimdf.3 . . . 4  |-  ( ph  ->  F/ x ps )
31, 2alrimi 1460 . . 3  |-  ( ph  ->  A. z F/ x ps )
4 nfsb4t 1938 . . 3  |-  ( A. z F/ x ps  ->  ( -.  A. x  x  =  y  ->  F/ x [ y  /  z ] ps ) )
53, 4syl 14 . 2  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x [ y  / 
z ] ps )
)
6 dvelimdf.1 . . 3  |-  F/ x ph
7 dvelimdf.4 . . . 4  |-  ( ph  ->  F/ z ch )
8 dvelimdf.5 . . . 4  |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch )
) )
91, 7, 8sbied 1718 . . 3  |-  ( ph  ->  ( [ y  / 
z ] ps  <->  ch )
)
106, 9nfbidf 1477 . 2  |-  ( ph  ->  ( F/ x [
y  /  z ] ps  <->  F/ x ch )
)
115, 10sylibd 147 1  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103   A.wal 1287   F/wnf 1394   [wsb 1692
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693
This theorem is referenced by:  dvelimdc  2248
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