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Theorem eliunid 42159
 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 3229 . 2 ((𝑥𝐴𝐶𝐵) → ∃𝑥𝐴 𝐶𝐵)
2 eliun 4888 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝐶𝐵)
31, 2sylibr 237 1 ((𝑥𝐴𝐶𝐵) → 𝐶 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2112  ∃wrex 3072  ∪ ciun 4884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-v 3412  df-iun 4886 This theorem is referenced by: (None)
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