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Theorem hbsb4 1965
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 1900 . 2  |-  ( [ w  /  x ] ph  ->  A. z [ w  /  x ] ph )
3 sbequ 1796 . 2  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
42, 3dvelimALT 1963 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 1314   [wsb 1720
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-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721
This theorem is referenced by:  hbsb4t  1966  dvelimf  1968
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