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Theorem sbceqal 3030
Description: A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
Assertion
Ref Expression
sbceqal  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
Distinct variable groups:    x, B    x, A
Allowed substitution hint:    V( x)

Proof of Theorem sbceqal
StepHypRef Expression
1 spsbc 2986 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  [. A  /  x ]. ( x  =  A  ->  x  =  B ) ) )
2 sbcimg 3016 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( x  =  A  ->  x  =  B )  <->  ( [. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B ) ) )
3 eqid 2187 . . . . 5  |-  A  =  A
4 eqsbc1 3014 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
53, 4mpbiri 168 . . . 4  |-  ( A  e.  V  ->  [. A  /  x ]. x  =  A )
6 pm5.5 242 . . . 4  |-  ( [. A  /  x ]. x  =  A  ->  ( (
[. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B
)  <->  [. A  /  x ]. x  =  B
) )
75, 6syl 14 . . 3  |-  ( A  e.  V  ->  (
( [. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B
)  <->  [. A  /  x ]. x  =  B
) )
8 eqsbc1 3014 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
92, 7, 83bitrd 214 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( x  =  A  ->  x  =  B )  <->  A  =  B
) )
101, 9sylibd 149 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1361    = wceq 1363    e. wcel 2158   [.wsbc 2974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975
This theorem is referenced by:  sbeqalb  3031  snsssn  3773
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