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Theorem eliuni 4748
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 2892 . . 3 (𝑥 = 𝐴 → (𝐸𝐵𝐸𝐶))
32rspcev 3526 . 2 ((𝐴𝐷𝐸𝐶) → ∃𝑥𝐷 𝐸𝐵)
4 eliun 4746 . 2 (𝐸 𝑥𝐷 𝐵 ↔ ∃𝑥𝐷 𝐸𝐵)
53, 4sylibr 226 1 ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wrex 3118   ciun 4742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-iun 4744
This theorem is referenced by:  oeordi  7939  fseqdom  9169  cfsmolem  9414  axdc3lem2  9595  prmreclem5  16002  efgs1b  18507  lbsextlem2  19527  pmatcoe1fsupp  20883  vitalilem2  23782  cnrefiisplem  40844
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