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Theorem sbal1 1977
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 1975 . . . 4  |-  ( [ w  /  y ] A. x ph  <->  A. x [ w  /  y ] ph )
21sbbii 1738 . . 3  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  [ z  /  w ] A. x [ w  /  y ] ph )
3 sbal1yz 1976 . . 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 1506 . . 3  |-  ( A. x ph  ->  A. w A. x ph )
65sbco2vh 1918 . 2  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  [ z  /  y ] A. x ph )
7 ax-17 1506 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2vh 1918 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98albii 1446 . 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 1329   [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-in2 604  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)
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