Theorem difindi 4233

Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difindi	⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))

Proof of Theorem difindi

Step	Hyp	Ref	Expression
1		dfin3 4218	. . 3 ⊢ (𝐵 ∩ 𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))
2	1	difeq2i 4064	. 2 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
3		indi 4225	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶)))
4		dfin2 4212	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
5		invdif 4220	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)
6		invdif 4220	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
7	5, 6	uneq12i 4107	. . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))
8	3, 4, 7	3eqtr3i 2768	. 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))
9	2, 8	eqtri 2760	1 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))

Syntax hints:   = wceq 1542  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897

This theorem is referenced by:  difdif2  4237  indm  4239  fndifnfp  7126  dprddisj2  20011  fctop  22983  cctop  22985  mretopd  23071  restcld  23151  cfinfil  23872  csdfil  23873  indifundif  32613  difres  32689  unelcarsg  34476  clsk3nimkb  44489  ntrclskb  44518  ntrclsk3  44519  ntrclsk13  44520  salincl  46774  iscnrm3rlem1  49431