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Theorem iunconstm 3690
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 3338 . . 3 (∃𝑥 𝑥𝐴 → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
2 eliun 3686 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
31, 2syl6rbbr 192 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2052 1 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1257  wex 1395  wcel 1407  wrex 2322   ciun 3682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-iun 3684
This theorem is referenced by: (None)
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