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Theorem sbceqal 2878
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 2835 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  [. A  /  x ]. ( x  =  A  ->  x  =  B ) ) )
2 sbcimg 2864 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( x  =  A  ->  x  =  B )  <->  ( [. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B ) ) )
3 eqid 2083 . . . . 5  |-  A  =  A
4 eqsbc3 2862 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
53, 4mpbiri 166 . . . 4  |-  ( A  e.  V  ->  [. A  /  x ]. x  =  A )
6 pm5.5 240 . . . 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 eqsbc3 2862 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
92, 7, 83bitrd 212 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( x  =  A  ->  x  =  B )  <->  A  =  B
) )
101, 9sylibd 147 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   [.wsbc 2824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by:  sbeqalb  2879  snsssn  3573
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