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Theorem eqsbc3 3366
Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2582. (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 3328 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. x  =  B  <->  [. A  /  x ]. x  =  B )
)
2 eqeq1 2466 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
3 sbsbc 3330 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  [. y  /  x ]. x  =  B )
4 eqsb3 2582 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  y  =  B )
53, 4bitr3i 251 . 2  |-  ( [. y  /  x ]. x  =  B  <->  y  =  B )
61, 2, 5vtoclbg 3167 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374   [wsb 1706    e. wcel 1762   [.wsbc 3326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3110  df-sbc 3327
This theorem is referenced by:  sbceqal  3382  eqsbc3r  3388  fmptsnd  6076  fvmptnn04if  19112  snfil  20095  iotavalb  30872  onfrALTlem5  32271  eqsbc3rVD  32597  onfrALTlem5VD  32642
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