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Theorem equsb3 1944
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
Assertion
Ref Expression
equsb3  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
Distinct variable group:    x, z

Proof of Theorem equsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 1943 . . 3  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
21sbbii 1758 . 2  |-  ( [ y  /  w ] [ w  /  x ] x  =  z  <->  [ y  /  w ]
w  =  z )
3 ax-17 1519 . . 3  |-  ( x  =  z  ->  A. w  x  =  z )
43sbco2vh 1938 . 2  |-  ( [ y  /  w ] [ w  /  x ] x  =  z  <->  [ y  /  x ]
x  =  z )
5 equsb3lem 1943 . 2  |-  ( [ y  /  w ]
w  =  z  <->  y  =  z )
62, 4, 53bitr3i 209 1  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1755
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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756
This theorem is referenced by:  sb8eu  2032  sb8euh  2042  sb8iota  5165
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