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Theorem iniin2 40230
 Description: Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
iniin2 (𝐴 ≠ ∅ → (𝐵 𝑥𝐴 𝐶) = 𝑥𝐴 (𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iniin2
StepHypRef Expression
1 iinin2 4823 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
21eqcomd 2783 1 (𝐴 ≠ ∅ → (𝐵 𝑥𝐴 𝐶) = 𝑥𝐴 (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ≠ wne 2968   ∩ cin 3790  ∅c0 4140  ∩ ciin 4754 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-v 3399  df-dif 3794  df-in 3798  df-nul 4141  df-iin 4756 This theorem is referenced by: (None)
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