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Theorem difun1 3920
 Description: A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
Assertion
Ref Expression
difun1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Proof of Theorem difun1
StepHypRef Expression
1 inass 3856 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
2 invdif 3901 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
31, 2eqtr3i 2675 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
4 undm 3918 . . . . 5 (V ∖ (𝐵𝐶)) = ((V ∖ 𝐵) ∩ (V ∖ 𝐶))
54ineq2i 3844 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
6 invdif 3901 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∖ (𝐵𝐶))
75, 6eqtr3i 2675 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (𝐵𝐶))
83, 7eqtr3i 2675 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = (𝐴 ∖ (𝐵𝐶))
9 invdif 3901 . . 3 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
109difeq1i 3757 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = ((𝐴𝐵) ∖ 𝐶)
118, 10eqtr3i 2675 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
 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:  dif32  3924  difabs  3925  difpr  4366  infdiffi  8593  mreexexlem4d  16354  nulmbl2  23350  unmbl  23351  caragenuncllem  41047
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