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Theorem iunconstm 3665
Description: Indexed union of a constant class, i.e. where  B does not depend on  x. (Contributed by Jim Kingdon, 15-Aug-2018.)
Assertion
Ref Expression
iunconstm  |-  ( E. x  x  e.  A  ->  U_ x  e.  A  B  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem iunconstm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.9rmv 3313 . . 3  |-  ( E. x  x  e.  A  ->  ( y  e.  B  <->  E. x  e.  A  y  e.  B ) )
2 eliun 3661 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
31, 2syl6rbbr 188 . 2  |-  ( E. x  x  e.  A  ->  ( y  e.  U_ x  e.  A  B  <->  y  e.  B ) )
43eqrdv 2038 1  |-  ( E. x  x  e.  A  ->  U_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   E.wex 1381    e. wcel 1393   E.wrex 2307   U_ciun 3657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-iun 3659
This theorem is referenced by: (None)
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