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Theorem inindi 3791
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 3783 . . 3 (𝐴𝐴) = 𝐴
21ineq1i 3771 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵𝐶))
3 in4 3790 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
42, 3eqtr3i 2633 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  cin 3538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-in 3546
This theorem is referenced by:  difundi  3837  dfif5  4051  resindi  5319  offres  7031  incexclem  14353  bitsinv1  14948  bitsinvp1  14955  bitsres  14979  fh1  27667
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