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Theorem inindi 3961
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 3953 . . 3 (𝐴𝐴) = 𝐴
21ineq1i 3941 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵𝐶))
3 in4 3960 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
42, 3eqtr3i 2772 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1620  cin 3702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-v 3330  df-in 3710
This theorem is referenced by:  difundi  4010  dfif5  4234  resindi  5558  offres  7316  incexclem  14738  bitsinv1  15337  bitsinvp1  15344  bitsres  15368  fh1  28757
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