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Theorem eqsbc3 3367
Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2577. (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 3329 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. x  =  B  <->  [. A  /  x ]. x  =  B )
)
2 eqeq1 2461 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
3 sbsbc 3331 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  [. y  /  x ]. x  =  B )
4 eqsb3 2577 . . 3  |-  ( [ y  /  x ]
x  =  B  <->  y  =  B )
53, 4bitr3i 251 . 2  |-  ( [. y  /  x ]. x  =  B  <->  y  =  B )
61, 2, 5vtoclbg 3168 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395   [wsb 1740    e. wcel 1819   [.wsbc 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328
This theorem is referenced by:  sbceqal  3383  eqsbc3r  3389  fmptsnd  6094  fvmptnn04if  19476  snfil  20490  iotavalb  31499  onfrALTlem5  33415  eqsbc3rVD  33741  onfrALTlem5VD  33786
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