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Theorem in12 3967
 Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
Assertion
Ref Expression
in12 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))

Proof of Theorem in12
StepHypRef Expression
1 incom 3948 . . 3 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 3953 . 2 ((𝐴𝐵) ∩ 𝐶) = ((𝐵𝐴) ∩ 𝐶)
3 inass 3966 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
4 inass 3966 . 2 ((𝐵𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴𝐶))
52, 3, 43eqtr3i 2790 1 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∩ cin 3714 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-v 3342  df-in 3722 This theorem is referenced by:  in32  3968  in31  3970  in4  3972  resdmres  5786  kmlem12  9175  ressress  16140  fh1  28786  fh2  28787  mdslmd3i  29500  bj-inrab3  33231
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