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Theorem nfsbt 1892
Description: Closed form of nfsb 1864. (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 1460 . 2  |-  ( A. x F/ z ph  ->  A. w A. x F/ z ph )
2 nfsbxyt 1861 . . . . 5  |-  ( A. x F/ z ph  ->  F/ z [ w  /  x ] ph )
32alimi 1385 . . . 4  |-  ( A. w A. x F/ z
ph  ->  A. w F/ z [ w  /  x ] ph )
4 nfsbxyt 1861 . . . 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 1462 . . . . 5  |-  F/ w ph
76sbco2 1881 . . . 4  |-  ( [ y  /  w ] [ w  /  x ] ph  <->  [ y  /  x ] ph )
87nfbii 1403 . . 3  |-  ( F/ z [ y  /  w ] [ w  /  x ] ph  <->  F/ z [ y  /  x ] ph )
95, 8sylib 120 . 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 1283   F/wnf 1390   [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-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:  nfsbd  1893  setindft  10918
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