MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difdif2 Structured version   Visualization version   GIF version

Theorem difdif2 3917
Description: Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
Assertion
Ref Expression
difdif2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem difdif2
StepHypRef Expression
1 difindi 3914 . 2 (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2 invdif 3901 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
32eqcomi 2660 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
43difeq2i 3758 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5 dfin2 3893 . . 3 (𝐴𝐶) = (𝐴 ∖ (V ∖ 𝐶))
65uneq2i 3797 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
71, 4, 63eqtr4i 2683 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  Vcvv 3231  cdif 3604  cun 3605  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-un 3612  df-in 3614
This theorem is referenced by:  restmetu  22422  difelcarsg  30500  mblfinlem3  33578  mblfinlem4  33579
  Copyright terms: Public domain W3C validator