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Theorem eqsb3 2221
Description: Substitution applied to an atomic wff (class version of equsb3 1902). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3  |-  ( [ y  /  x ]
x  =  A  <->  y  =  A )
Distinct variable group:    x, A
Allowed substitution hint:    A( y)

Proof of Theorem eqsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eqsb3lem 2220 . . 3  |-  ( [ w  /  x ]
x  =  A  <->  w  =  A )
21sbbii 1723 . 2  |-  ( [ y  /  w ] [ w  /  x ] x  =  A  <->  [ y  /  w ]
w  =  A )
3 nfv 1493 . . 3  |-  F/ w  x  =  A
43sbco2 1916 . 2  |-  ( [ y  /  w ] [ w  /  x ] x  =  A  <->  [ y  /  x ]
x  =  A )
5 eqsb3lem 2220 . 2  |-  ( [ y  /  w ]
w  =  A  <->  y  =  A )
62, 4, 53bitr3i 209 1  |-  ( [ y  /  x ]
x  =  A  <->  y  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1316   [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-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  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-cleq 2110
This theorem is referenced by:  pm13.183  2796  eqsbc3  2920
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