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Theorem iunconstm 3906
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 3902 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 r19.9rmv 3526 . . 3 (∃𝑥 𝑥𝐴 → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
31, 2bitr4id 199 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2185 1 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wex 1502  wcel 2158  wrex 2466   ciun 3898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-iun 3900
This theorem is referenced by: (None)
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