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Theorem inindi 4187
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 4179 . . 3 (𝐴𝐴) = 𝐴
21ineq1i 4169 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵𝐶))
3 in4 4186 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
42, 3eqtr3i 2767 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cin 3910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-in 3918
This theorem is referenced by:  difundi  4240  dfif5  4503  resindi  5954  offres  7917  incexclem  15722  bitsinv1  16323  bitsinvp1  16330  bitsres  16354  fh1  30563
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