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Theorem bnj956 28865
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 2473 . . . . . . . 8  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
32anbi1d 686 . . . . . . 7  |-  ( A  =  B  ->  (
( x  e.  A  /\  y  e.  C
)  <->  ( x  e.  B  /\  y  e.  C ) ) )
43alimi 1565 . . . . . 6  |-  ( A. x  A  =  B  ->  A. x ( ( x  e.  A  /\  y  e.  C )  <->  ( x  e.  B  /\  y  e.  C )
) )
5 exbi 1588 . . . . . 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 2680 . . . . 5  |-  ( E. x  e.  A  y  e.  C  <->  E. x
( x  e.  A  /\  y  e.  C
) )
8 df-rex 2680 . . . . 5  |-  ( E. x  e.  B  y  e.  C  <->  E. x
( x  e.  B  /\  y  e.  C
) )
96, 7, 83bitr4g 280 . . . 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 2526 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  A  y  e.  C }  =  { y  |  E. x  e.  B  y  e.  C }
)
12 df-iun 4063 . 2  |-  U_ x  e.  A  C  =  { y  |  E. x  e.  A  y  e.  C }
13 df-iun 4063 . 2  |-  U_ x  e.  B  C  =  { y  |  E. x  e.  B  y  e.  C }
1411, 12, 133eqtr4g 2469 1  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2398   E.wrex 2675   U_ciun 4061
This theorem is referenced by:  bnj1316  28910  bnj953  29028  bnj1000  29030  bnj966  29033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-rex 2680  df-iun 4063
  Copyright terms: Public domain W3C validator