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Theorem hbsb4 2022
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 1959 . 2  |-  ( [ w  /  x ] ph  ->  A. z [ w  /  x ] ph )
3 sbequ 1850 . 2  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
42, 3dvelimALT 2020 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 1361   [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-in2 616  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:  hbsb4t  2023  dvelimf  2025
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