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Theorem eliuni 4658
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 2835 . . 3 (𝑥 = 𝐴 → (𝐸𝐵𝐸𝐶))
32rspcev 3458 . 2 ((𝐴𝐷𝐸𝐶) → ∃𝑥𝐷 𝐸𝐵)
4 eliun 4656 . 2 (𝐸 𝑥𝐷 𝐵 ↔ ∃𝑥𝐷 𝐸𝐵)
53, 4sylibr 224 1 ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wrex 3061   ciun 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-iun 4654
This theorem is referenced by:  oeordi  7820  fseqdom  9048  cfsmolem  9293  axdc3lem2  9474  prmreclem5  15830  efgs1b  18355  lbsextlem2  19373  pmatcoe1fsupp  20725  vitalilem2  23596  cnrefiisplem  40567
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