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Theorem difdif2 4033
 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 4030 . 2 (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2 invdif 4017 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
32eqcomi 2780 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
43difeq2i 3876 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5 dfin2 4009 . . 3 (𝐴𝐶) = (𝐴 ∖ (V ∖ 𝐶))
65uneq2i 3915 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
71, 4, 63eqtr4i 2803 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631  Vcvv 3351   ∖ cdif 3720   ∪ cun 3721   ∩ cin 3722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730 This theorem is referenced by:  restmetu  22595  difelcarsg  30712  mblfinlem3  33781  mblfinlem4  33782
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