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Theorem eqsbc3 2802
Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2141. (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 2766 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. x  =  B  <->  [. A  /  x ]. x  =  B )
)
2 eqeq1 2046 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
3 sbsbc 2768 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  [. y  /  x ]. x  =  B )
4 eqsb3 2141 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  y  =  B )
53, 4bitr3i 175 . 2  |-  ( [. y  /  x ]. x  =  B  <->  y  =  B )
61, 2, 5vtoclbg 2614 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    e. wcel 1393   [wsb 1645   [.wsbc 2764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765
This theorem is referenced by:  sbceqal  2814  eqsbc3r  2819
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