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Theorem bnj956 32789
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 2533 . . . . . . . 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 1609 . . . . . 6  |-  ( A. x  A  =  B  ->  A. x ( ( x  e.  A  /\  y  e.  C )  <->  ( x  e.  B  /\  y  e.  C )
) )
5 exbi 1638 . . . . . 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 2813 . . . . 5  |-  ( E. x  e.  A  y  e.  C  <->  E. x
( x  e.  A  /\  y  e.  C
) )
8 df-rex 2813 . . . . 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 2596 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  A  y  e.  C }  =  { y  |  E. x  e.  B  y  e.  C }
)
12 df-iun 4320 . 2  |-  U_ x  e.  A  C  =  { y  |  E. x  e.  A  y  e.  C }
13 df-iun 4320 . 2  |-  U_ x  e.  B  C  =  { y  |  E. x  e.  B  y  e.  C }
1411, 12, 133eqtr4g 2526 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 1372    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2445   E.wrex 2808   U_ciun 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-rex 2813  df-iun 4320
This theorem is referenced by:  bnj1316  32833  bnj953  32951  bnj1000  32953  bnj966  32956
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