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Theorem eqsbc3 3200
Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2537. (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 3163 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. x  =  B  <->  [. A  /  x ]. x  =  B )
)
2 eqeq1 2442 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
3 sbsbc 3165 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  [. y  /  x ]. x  =  B )
4 eqsb3 2537 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  y  =  B )
53, 4bitr3i 243 . 2  |-  ( [. y  /  x ]. x  =  B  <->  y  =  B )
61, 2, 5vtoclbg 3012 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   [wsb 1658    e. wcel 1725   [.wsbc 3161
This theorem is referenced by:  sbceqal  3212  eqsbc3r  3218  snfil  17896  iotavalb  27607  onfrALTlem5  28628  eqsbc3rVD  28952  onfrALTlem5VD  28997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162
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