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Theorem eliunid 44395
Description: Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Assertion
Ref Expression
eliunid ((𝑥𝐴𝐶𝐵) → 𝐶 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem eliunid
StepHypRef Expression
1 rspe 3240 . 2 ((𝑥𝐴𝐶𝐵) → ∃𝑥𝐴 𝐶𝐵)
2 eliun 4994 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝐶𝐵)
31, 2sylibr 233 1 ((𝑥𝐴𝐶𝐵) → 𝐶 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wrex 3064   ciun 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rex 3065  df-v 3470  df-iun 4992
This theorem is referenced by: (None)
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