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Theorem iniin1 39830
 Description: Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
iniin1 (𝐴 ≠ ∅ → ( 𝑥𝐴 𝐶𝐵) = 𝑥𝐴 (𝐶𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iniin1
StepHypRef Expression
1 iinin1 4725 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
21eqcomd 2777 1 (𝐴 ≠ ∅ → ( 𝑥𝐴 𝐶𝐵) = 𝑥𝐴 (𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ≠ wne 2943   ∩ cin 3722  ∅c0 4063  ∩ ciin 4655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064  df-iin 4657 This theorem is referenced by:  smfsuplem1  41537
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