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Theorem sbceqal 3042
Description: Set theory version of sbeqal1 27597. (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 3003 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  [. A  /  x ]. ( x  =  A  ->  x  =  B ) ) )
2 sbcimg 3032 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( x  =  A  ->  x  =  B )  <->  ( [. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B ) ) )
3 eqid 2283 . . . . 5  |-  A  =  A
4 eqsbc3 3030 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
53, 4mpbiri 224 . . . 4  |-  ( A  e.  V  ->  [. A  /  x ]. x  =  A )
6 pm5.5 326 . . . 4  |-  ( [. A  /  x ]. x  =  A  ->  ( (
[. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B
)  <->  [. A  /  x ]. x  =  B
) )
75, 6syl 15 . . 3  |-  ( A  e.  V  ->  (
( [. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B
)  <->  [. A  /  x ]. x  =  B
) )
8 eqsbc3 3030 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
92, 7, 83bitrd 270 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( x  =  A  ->  x  =  B )  <->  A  =  B
) )
101, 9sylibd 205 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  sbeqalb  3043
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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