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Theorem difdif2 4235
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 4232 . 2 (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2 invdif 4219 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
32eqcomi 2831 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
43difeq2i 4071 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5 dfin2 4211 . . 3 (𝐴𝐶) = (𝐴 ∖ (V ∖ 𝐶))
65uneq2i 4111 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
71, 4, 63eqtr4i 2855 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  Vcvv 3469  cdif 3905  cun 3906  cin 3907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915
This theorem is referenced by:  restmetu  23175  difelcarsg  31642  mblfinlem3  35054  mblfinlem4  35055
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