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Theorem nfsbt 1949
Description: Closed form of nfsb 1919. (Contributed by Jim Kingdon, 9-May-2018.)
Assertion
Ref Expression
nfsbt  |-  ( A. x F/ z ph  ->  F/ z [ y  /  x ] ph )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem nfsbt
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1506 . 2  |-  ( A. x F/ z ph  ->  A. w A. x F/ z ph )
2 nfsbxyt 1916 . . . . 5  |-  ( A. x F/ z ph  ->  F/ z [ w  /  x ] ph )
32alimi 1431 . . . 4  |-  ( A. w A. x F/ z
ph  ->  A. w F/ z [ w  /  x ] ph )
4 nfsbxyt 1916 . . . 4  |-  ( A. w F/ z [ w  /  x ] ph  ->  F/ z [ y  /  w ] [ w  /  x ] ph )
53, 4syl 14 . . 3  |-  ( A. w A. x F/ z
ph  ->  F/ z [ y  /  w ] [ w  /  x ] ph )
6 nfv 1508 . . . . 5  |-  F/ w ph
76sbco2 1938 . . . 4  |-  ( [ y  /  w ] [ w  /  x ] ph  <->  [ y  /  x ] ph )
87nfbii 1449 . . 3  |-  ( F/ z [ y  /  w ] [ w  /  x ] ph  <->  F/ z [ y  /  x ] ph )
95, 8sylib 121 . 2  |-  ( A. w A. x F/ z
ph  ->  F/ z [ y  /  x ] ph )
101, 9syl 14 1  |-  ( A. x F/ z ph  ->  F/ z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1329   F/wnf 1436   [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  nfsbd  1950  setindft  13177
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