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Theorem nfeudv 1963
Description: Deduction version of nfeu 1967. Similar to nfeud 1964 but has the additional constraint that  x and  y must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)
Hypotheses
Ref Expression
nfeudv.1  |-  F/ y
ph
nfeudv.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfeudv  |-  ( ph  ->  F/ x E! y ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem nfeudv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1466 . . 3  |-  F/ z
ph
2 nfeudv.1 . . . 4  |-  F/ y
ph
3 nfeudv.2 . . . . 5  |-  ( ph  ->  F/ x ps )
4 nfv 1466 . . . . . 6  |-  F/ x  y  =  z
54a1i 9 . . . . 5  |-  ( ph  ->  F/ x  y  =  z )
63, 5nfbid 1525 . . . 4  |-  ( ph  ->  F/ x ( ps  <->  y  =  z ) )
72, 6nfald 1690 . . 3  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
81, 7nfexd 1691 . 2  |-  ( ph  ->  F/ x E. z A. y ( ps  <->  y  =  z ) )
9 df-eu 1951 . . 3  |-  ( E! y ps  <->  E. z A. y ( ps  <->  y  =  z ) )
109nfbii 1407 . 2  |-  ( F/ x E! y ps  <->  F/ x E. z A. y ( ps  <->  y  =  z ) )
118, 10sylibr 132 1  |-  ( ph  ->  F/ x E! y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289   F/wnf 1394   E.wex 1426   E!weu 1948
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-eu 1951
This theorem is referenced by:  nfeud  1964
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