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Theorem iunconstm 3639
 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 (x x A x 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 r19.9rmvOLD 3292 . . 3 (x x A → (y Bx A y B))
2 eliun 3635 . . 3 (y x A Bx A y B)
31, 2syl6rbbr 188 . 2 (x x A → (y x A By B))
43eqrdv 2020 1 (x x A x A B = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃wrex 2285  ∪ ciun 3631 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-iun 3633 This theorem is referenced by: (None)
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