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Theorem bnj956 32072
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj956.1  |-  ( A  =  B  ->  A. x  A  =  B )
Assertion
Ref Expression
bnj956  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)

Proof of Theorem bnj956
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bnj956.1 . . . 4  |-  ( A  =  B  ->  A. x  A  =  B )
2 eleq2 2524 . . . . . . . 8  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
32anbi1d 704 . . . . . . 7  |-  ( A  =  B  ->  (
( x  e.  A  /\  y  e.  C
)  <->  ( x  e.  B  /\  y  e.  C ) ) )
43alimi 1605 . . . . . 6  |-  ( A. x  A  =  B  ->  A. x ( ( x  e.  A  /\  y  e.  C )  <->  ( x  e.  B  /\  y  e.  C )
) )
5 exbi 1634 . . . . . 6  |-  ( A. x ( ( x  e.  A  /\  y  e.  C )  <->  ( x  e.  B  /\  y  e.  C ) )  -> 
( E. x ( x  e.  A  /\  y  e.  C )  <->  E. x ( x  e.  B  /\  y  e.  C ) ) )
64, 5syl 16 . . . . 5  |-  ( A. x  A  =  B  ->  ( E. x ( x  e.  A  /\  y  e.  C )  <->  E. x ( x  e.  B  /\  y  e.  C ) ) )
7 df-rex 2801 . . . . 5  |-  ( E. x  e.  A  y  e.  C  <->  E. x
( x  e.  A  /\  y  e.  C
) )
8 df-rex 2801 . . . . 5  |-  ( E. x  e.  B  y  e.  C  <->  E. x
( x  e.  B  /\  y  e.  C
) )
96, 7, 83bitr4g 288 . . . 4  |-  ( A. x  A  =  B  ->  ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  C ) )
101, 9syl 16 . . 3  |-  ( A  =  B  ->  ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  C ) )
1110abbidv 2587 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  A  y  e.  C }  =  { y  |  E. x  e.  B  y  e.  C }
)
12 df-iun 4273 . 2  |-  U_ x  e.  A  C  =  { y  |  E. x  e.  A  y  e.  C }
13 df-iun 4273 . 2  |-  U_ x  e.  B  C  =  { y  |  E. x  e.  B  y  e.  C }
1411, 12, 133eqtr4g 2517 1  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   E.wrex 2796   U_ciun 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-rex 2801  df-iun 4273
This theorem is referenced by:  bnj1316  32116  bnj953  32234  bnj1000  32236  bnj966  32239
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