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Theorem eqsbc3 2920
Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2221. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
eqsbc3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    V( x)

Proof of Theorem eqsbc3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2884 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. x  =  B  <->  [. A  /  x ]. x  =  B )
)
2 eqeq1 2124 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
3 sbsbc 2886 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  [. y  /  x ]. x  =  B )
4 eqsb3 2221 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  y  =  B )
53, 4bitr3i 185 . 2  |-  ( [. y  /  x ]. x  =  B  <->  y  =  B )
61, 2, 5vtoclbg 2721 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   [wsb 1720   [.wsbc 2882
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-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by:  sbceqal  2936  eqsbc3r  2941
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