ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfsb4or Unicode version

Theorem nfsb4or 1998
Description: A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)
Hypothesis
Ref Expression
nfsb4or.1  |-  F/ z
ph
Assertion
Ref Expression
nfsb4or  |-  ( A. z  z  =  y  \/  F/ z [ y  /  x ] ph )

Proof of Theorem nfsb4or
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfsb4or.1 . . 3  |-  F/ z
ph
21nfsb 1919 . 2  |-  F/ z [ w  /  x ] ph
3 sbequ 1812 . 2  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
42, 3dvelimor 1993 1  |-  ( A. z  z  =  y  \/  F/ z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    \/ wo 697   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: (None)
  Copyright terms: Public domain W3C validator