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Theorem inindi 3792
 Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
Assertion
Ref Expression
inindi (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem inindi
StepHypRef Expression
1 inidm 3784 . . 3 (𝐴𝐴) = 𝐴
21ineq1i 3772 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵𝐶))
3 in4 3791 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
42, 3eqtr3i 2634 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547 This theorem is referenced by:  difundi  3838  dfif5  4052  resindi  5332  offres  7054  incexclem  14407  bitsinv1  15002  bitsinvp1  15009  bitsres  15033  fh1  27861
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