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Theorem eliuni 4918
 Description: Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)
Hypothesis
Ref Expression
eliuni.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
eliuni ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eliuni
StepHypRef Expression
1 eliuni.1 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
21eleq2d 2898 . . 3 (𝑥 = 𝐴 → (𝐸𝐵𝐸𝐶))
32rspcev 3623 . 2 ((𝐴𝐷𝐸𝐶) → ∃𝑥𝐷 𝐸𝐵)
4 eliun 4916 . 2 (𝐸 𝑥𝐷 𝐵 ↔ ∃𝑥𝐷 𝐸𝐵)
53, 4sylibr 236 1 ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  ∪ ciun 4912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3497  df-iun 4914 This theorem is referenced by:  oeordi  8207  fseqdom  9446  cfsmolem  9686  axdc3lem2  9867  prmreclem5  16250  efgs1b  18856  lbsextlem2  19925  pmatcoe1fsupp  21303  vitalilem2  24204  cnrefiisplem  42102
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