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Theorem equsb3 1867
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
Assertion
Ref Expression
equsb3  |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
Distinct variable group:    y, z

Proof of Theorem equsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 1866 . . 3  |-  ( [ w  /  y ] y  =  z  <->  w  =  z )
21sbbii 1689 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] w  =  z
)
3 ax-17 1460 . . 3  |-  ( y  =  z  ->  A. w  y  =  z )
43sbco2v 1863 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
5 equsb3lem 1866 . 2  |-  ( [ x  /  w ]
w  =  z  <->  x  =  z )
62, 4, 53bitr3i 208 1  |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  sb8eu  1955  sb8euh  1965  sb8iota  4904
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