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Theorem eliuni 4492
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 2684 . . 3 (𝑥 = 𝐴 → (𝐸𝐵𝐸𝐶))
32rspcev 3295 . 2 ((𝐴𝐷𝐸𝐶) → ∃𝑥𝐷 𝐸𝐵)
4 eliun 4490 . 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 384   = wceq 1480  wcel 1987  wrex 2908   ciun 4485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-iun 4487
This theorem is referenced by:  oeordi  7612  fseqdom  8793  cfsmolem  9036  axdc3lem2  9217  prmreclem5  15548  efgs1b  18070  lbsextlem2  19078  pmatcoe1fsupp  20425  vitalilem2  23284
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