MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbceqal Structured version   Visualization version   Unicode version

Theorem sbceqal 3330
Description: Set theory version of sbeqal1 36791. (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 3291 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  [. A  /  x ]. ( x  =  A  ->  x  =  B ) ) )
2 sbcimg 3320 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( x  =  A  ->  x  =  B )  <->  ( [. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B ) ) )
3 eqid 2461 . . . . 5  |-  A  =  A
4 eqsbc3 3318 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
53, 4mpbiri 241 . . . 4  |-  ( A  e.  V  ->  [. A  /  x ]. x  =  A )
6 pm5.5 342 . . . 4  |-  ( [. A  /  x ]. x  =  A  ->  ( (
[. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B
)  <->  [. A  /  x ]. x  =  B
) )
75, 6syl 17 . . 3  |-  ( A  e.  V  ->  (
( [. A  /  x ]. x  =  A  ->  [. A  /  x ]. x  =  B
)  <->  [. A  /  x ]. x  =  B
) )
8 eqsbc3 3318 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
92, 7, 83bitrd 287 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( x  =  A  ->  x  =  B )  <->  A  =  B
) )
101, 9sylibd 222 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1452    = wceq 1454    e. wcel 1897   [.wsbc 3278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3058  df-sbc 3279
This theorem is referenced by:  sbeqalb  3331
  Copyright terms: Public domain W3C validator