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Theorem iinss1 4469
 Description: Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)
Assertion
Ref Expression
iinss1 (𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iinss1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssralv 3629 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝑦𝐶 → ∀𝑥𝐴 𝑦𝐶))
2 vex 3176 . . . 4 𝑦 ∈ V
3 eliin 4461 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶))
42, 3ax-mp 5 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶)
5 eliin 4461 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
62, 5ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
71, 4, 63imtr4g 284 . 2 (𝐴𝐵 → (𝑦 𝑥𝐵 𝐶𝑦 𝑥𝐴 𝐶))
87ssrdv 3574 1 (𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∩ 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-in 3547  df-ss 3554  df-iin 4458 This theorem is referenced by:  polcon3N  34221
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