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Theorem eliind 38266
 Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
eliind.a (𝜑𝐴 𝑥𝐵 𝐶)
eliind.k (𝜑𝐾𝐵)
eliind.d (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))
Assertion
Ref Expression
eliind (𝜑𝐴𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐾
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem eliind
StepHypRef Expression
1 eliind.k . 2 (𝜑𝐾𝐵)
2 eliind.a . . 3 (𝜑𝐴 𝑥𝐵 𝐶)
3 eliin 4461 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
42, 3syl 17 . . 3 (𝜑 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
52, 4mpbid 221 . 2 (𝜑 → ∀𝑥𝐵 𝐴𝐶)
6 eliind.d . . 3 (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))
76rspcva 3280 . 2 ((𝐾𝐵 ∧ ∀𝑥𝐵 𝐴𝐶) → 𝐴𝐷)
81, 5, 7syl2anc 691 1 (𝜑𝐴𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ 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:  iooiinioc  38630  hspdifhsp  39506  smflimlem3  39659
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