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Theorem eliund 4998
Description: Membership in indexed union. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
Hypothesis
Ref Expression
eliund.1 (𝜑 → ∃𝑥𝐵 𝐴𝐶)
Assertion
Ref Expression
eliund (𝜑𝐴 𝑥𝐵 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliund
StepHypRef Expression
1 eliund.1 . 2 (𝜑 → ∃𝑥𝐵 𝐴𝐶)
2 eliun 4995 . 2 (𝐴 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝐴𝐶)
31, 2sylibr 234 1 (𝜑𝐴 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wrex 3070   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-iun 4993
This theorem is referenced by:  gsumwrd2dccatlem  33069
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