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Theorem eliind 40052
 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 4747 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
42, 3syl 17 . . 3 (𝜑 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
52, 4mpbid 224 . 2 (𝜑 → ∀𝑥𝐵 𝐴𝐶)
6 eliind.d . . 3 (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))
76rspcva 3524 . 2 ((𝐾𝐵 ∧ ∀𝑥𝐵 𝐴𝐶) → 𝐴𝐷)
81, 5, 7syl2anc 579 1 (𝜑𝐴𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∩ ciin 4743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-v 3416  df-iin 4745 This theorem is referenced by:  iooiinioc  40572  hspdifhsp  41618  smflimlem3  41769  smfsuplem1  41805  smflimsuplem4  41817
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