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Theorem iunconstm 3768
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 𝑥𝐴 𝑥𝐴 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 3401 . . 3 (∃𝑥 𝑥𝐴 → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
2 eliun 3764 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
31, 2syl6rbbr 198 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2098 1 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wex 1436  wcel 1448  wrex 2376   ciun 3760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-iun 3762
This theorem is referenced by: (None)
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