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Theorem nfsbt 1986
Description: Closed form of nfsb 1956. (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 1536 . 2  |-  ( A. x F/ z ph  ->  A. w A. x F/ z ph )
2 nfsbxyt 1953 . . . . 5  |-  ( A. x F/ z ph  ->  F/ z [ w  /  x ] ph )
32alimi 1465 . . . 4  |-  ( A. w A. x F/ z
ph  ->  A. w F/ z [ w  /  x ] ph )
4 nfsbxyt 1953 . . . 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 1538 . . . . 5  |-  F/ w ph
76sbco2 1975 . . . 4  |-  ( [ y  /  w ] [ w  /  x ] ph  <->  [ y  /  x ] ph )
87nfbii 1483 . . 3  |-  ( F/ z [ y  /  w ] [ w  /  x ] ph  <->  F/ z [ y  /  x ] ph )
95, 8sylib 122 . 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 1361   F/wnf 1470   [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773
This theorem is referenced by:  nfsbd  1987  setindft  15013
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