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Theorem iniin2 39809
 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 4742 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
21eqcomd 2766 1 (𝐴 ≠ ∅ → (𝐵 𝑥𝐴 𝐶) = 𝑥𝐴 (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ≠ wne 2932   ∩ cin 3714  ∅c0 4058  ∩ ciin 4673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-v 3342  df-dif 3718  df-in 3722  df-nul 4059  df-iin 4675 This theorem is referenced by: (None)
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