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Theorem iniin1 41268
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 4992 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
21eqcomd 2824 1 (𝐴 ≠ ∅ → ( 𝑥𝐴 𝐶𝐵) = 𝑥𝐴 (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wne 3013  cin 3932  c0 4288   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-nul 4289  df-iin 4913
This theorem is referenced by:  smfsuplem1  42962
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