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Theorem iniin1 42133
 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 4966 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
21eqcomd 2764 1 (𝐴 ≠ ∅ → ( 𝑥𝐴 𝐶𝐵) = 𝑥𝐴 (𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ≠ wne 2951   ∩ cin 3857  ∅c0 4225  ∩ ciin 4884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rab 3079  df-v 3411  df-dif 3861  df-in 3865  df-nul 4226  df-iin 4886 This theorem is referenced by:  smfsuplem1  43808
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