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Theorem iunconstm 3853
 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 eliun 3849 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 r19.9rmv 3481 . . 3 (∃𝑥 𝑥𝐴 → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
31, 2bitr4id 198 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2152 1 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  ∃wex 1469   ∈ wcel 2125  ∃wrex 2433  ∪ ciun 3845 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-iun 3847 This theorem is referenced by: (None)
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