Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  indifcom Structured version   Visualization version   GIF version

Theorem indifcom 3905
 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 3838 . . 3 (𝐴𝐵) = (𝐵𝐴)
21difeq1i 3757 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐵𝐴) ∖ 𝐶)
3 indif2 3903 . 2 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
4 indif2 3903 . 2 (𝐵 ∩ (𝐴𝐶)) = ((𝐵𝐴) ∖ 𝐶)
52, 3, 43eqtr4i 2683 1 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523   ∖ cdif 3604   ∩ cin 3606 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-in 3614 This theorem is referenced by:  ufprim  21760  cmmbl  23348  unmbl  23351  volinun  23360  limciun  23703  caragenuncllem  41047
 Copyright terms: Public domain W3C validator