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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 (∃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 3313 . . 3 (∃𝑥 𝑥𝐴 → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
2 eliun 3661 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
31, 2syl6rbbr 188 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2038 1 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ∪ 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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