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Theorem eliinid 38325
 Description: Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
eliinid ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliinid
StepHypRef Expression
1 simpl 472 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴 𝑥𝐵 𝐶)
2 eliin 4461 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
31, 2syl 17 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
41, 3mpbid 221 . 2 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → ∀𝑥𝐵 𝐴𝐶)
5 simpr 476 . 2 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝑥𝐵)
6 rspa 2914 . 2 ((∀𝑥𝐵 𝐴𝐶𝑥𝐵) → 𝐴𝐶)
74, 5, 6syl2anc 691 1 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-iin 4458 This theorem is referenced by:  fnlimfvre  38741  smflimlem2  39658
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