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Theorem hbsb4 1930
Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
Hypothesis
Ref Expression
hbsb4.1  |-  ( ph  ->  A. z ph )
Assertion
Ref Expression
hbsb4  |-  ( -. 
A. z  z  =  y  ->  ( [
y  /  x ] ph  ->  A. z [ y  /  x ] ph ) )

Proof of Theorem hbsb4
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 hbsb4.1 . . 3  |-  ( ph  ->  A. z ph )
21hbsb 1865 . 2  |-  ( [ w  /  x ] ph  ->  A. z [ w  /  x ] ph )
3 sbequ 1762 . 2  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
42, 3dvelimALT 1928 1  |-  ( -. 
A. z  z  =  y  ->  ( [
y  /  x ] ph  ->  A. z [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1283   [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-in2 578  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:  hbsb4t  1931  dvelimf  1933
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