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Theorem indifcom 4289
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 4217 . . 3 (𝐴𝐵) = (𝐵𝐴)
21difeq1i 4132 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐵𝐴) ∖ 𝐶)
3 indif2 4287 . 2 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
4 indif2 4287 . 2 (𝐵 ∩ (𝐴𝐶)) = ((𝐵𝐴) ∖ 𝐶)
52, 3, 43eqtr4i 2773 1 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cdif 3960  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970
This theorem is referenced by:  ufprim  23933  cmmbl  25583  unmbl  25586  volinun  25595  limciun  25944  caragenuncllem  46468
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