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Theorem eliind 44435
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.d . 2 (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))
2 eliind.a . . 3 (𝜑𝐴 𝑥𝐵 𝐶)
3 eliin 5001 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
42, 3syl 17 . . 3 (𝜑 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
52, 4mpbid 231 . 2 (𝜑 → ∀𝑥𝐵 𝐴𝐶)
6 eliind.k . 2 (𝜑𝐾𝐵)
71, 5, 6rspcdva 3610 1 (𝜑𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  wral 3058   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-iin 4999
This theorem is referenced by:  iooiinioc  44941  hspdifhsp  46004  smflimlem3  46161  smfsuplem1  46199  smflimsuplem4  46211
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