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Theorem sbal1 1955
Description: A theorem used in elimination of disjoint variable conditions on  x ,  y by replacing it with a distinctor  -.  A. x x  =  z. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.)
Assertion
Ref Expression
sbal1  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbal1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 sbal 1953 . . . 4  |-  ( [ w  /  y ] A. x ph  <->  A. x [ w  /  y ] ph )
21sbbii 1723 . . 3  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  [ z  /  w ] A. x [ w  /  y ] ph )
3 sbal1yz 1954 . . 3  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  w ] A. x [ w  / 
y ] ph  <->  A. x [ z  /  w ] [ w  /  y ] ph ) )
42, 3syl5bb 191 . 2  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  w ] [ w  /  y ] A. x ph  <->  A. x [ z  /  w ] [ w  /  y ] ph ) )
5 ax-17 1491 . . 3  |-  ( A. x ph  ->  A. w A. x ph )
65sbco2v 1898 . 2  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  [ z  /  y ] A. x ph )
7 ax-17 1491 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2v 1898 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98albii 1431 . 2  |-  ( A. x [ z  /  w ] [ w  /  y ] ph  <->  A. x [ z  /  y ] ph )
104, 6, 93bitr3g 221 1  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104   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: (None)
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