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Theorem eliind 41323
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 4915 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
42, 3syl 17 . . 3 (𝜑 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
52, 4mpbid 234 . 2 (𝜑 → ∀𝑥𝐵 𝐴𝐶)
6 eliind.k . 2 (𝜑𝐾𝐵)
71, 5, 6rspcdva 3623 1 (𝜑𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1530  wcel 2107  wral 3136   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-iin 4913
This theorem is referenced by:  iooiinioc  41821  hspdifhsp  42888  smflimlem3  43039  smfsuplem1  43075  smflimsuplem4  43087
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