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Theorem indifcom 4247
 Description: Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
Assertion
Ref Expression
indifcom (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))

Proof of Theorem indifcom
StepHypRef Expression
1 incom 4176 . . 3 (𝐴𝐵) = (𝐵𝐴)
21difeq1i 4093 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐵𝐴) ∖ 𝐶)
3 indif2 4245 . 2 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
4 indif2 4245 . 2 (𝐵 ∩ (𝐴𝐶)) = ((𝐵𝐴) ∖ 𝐶)
52, 3, 43eqtr4i 2852 1 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530   ∖ cdif 3931   ∩ cin 3933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-in 3941 This theorem is referenced by:  ufprim  22509  cmmbl  24127  unmbl  24130  volinun  24139  limciun  24484  caragenuncllem  42779
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