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Theorem sbal1 2002
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 2000 . . . 4  |-  ( [ w  /  y ] A. x ph  <->  A. x [ w  /  y ] ph )
21sbbii 1765 . . 3  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  [ z  /  w ] A. x [ w  /  y ] ph )
3 sbal1yz 2001 . . 3  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  w ] A. x [ w  / 
y ] ph  <->  A. x [ z  /  w ] [ w  /  y ] ph ) )
42, 3bitrid 192 . 2  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  w ] [ w  /  y ] A. x ph  <->  A. x [ z  /  w ] [ w  /  y ] ph ) )
5 ax-17 1526 . . 3  |-  ( A. x ph  ->  A. w A. x ph )
65sbco2vh 1945 . 2  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  [ z  /  y ] A. x ph )
7 ax-17 1526 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2vh 1945 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98albii 1470 . 2  |-  ( A. x [ z  /  w ] [ w  /  y ] ph  <->  A. x [ z  /  y ] ph )
104, 6, 93bitr3g 222 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 105   A.wal 1351   [wsb 1762
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 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by: (None)
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