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Theorem eliind2 41691
 Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
eliind2.1 𝑥𝜑
eliind2.2 (𝜑𝐴𝑉)
eliind2.3 ((𝜑𝑥𝐵) → 𝐴𝐶)
Assertion
Ref Expression
eliind2 (𝜑𝐴 𝑥𝐵 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem eliind2
StepHypRef Expression
1 eliind2.1 . . 3 𝑥𝜑
2 eliind2.3 . . . 4 ((𝜑𝑥𝐵) → 𝐴𝐶)
32ex 416 . . 3 (𝜑 → (𝑥𝐵𝐴𝐶))
41, 3ralrimi 3210 . 2 (𝜑 → ∀𝑥𝐵 𝐴𝐶)
5 eliind2.2 . . 3 (𝜑𝐴𝑉)
6 eliin 4910 . . 3 (𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
75, 6syl 17 . 2 (𝜑 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
84, 7mpbird 260 1 (𝜑𝐴 𝑥𝐵 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  Ⅎwnf 1785   ∈ wcel 2115  ∀wral 3133  ∩ ciin 4906 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-iin 4908 This theorem is referenced by: (None)
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