Theorem difindir 4247

Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difindir	⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (𝐵 ∖ 𝐶))

Proof of Theorem difindir

Step	Hyp	Ref	Expression
1		inindir 4190	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∩ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4233	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
3		invdif 4233	. . 3 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
4		invdif 4233	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
5	3, 4	ineq12i 4172	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐶)) ∩ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∩ (𝐵 ∖ 𝐶))
6	1, 2, 5	3eqtr3i 2768	1 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (𝐵 ∖ 𝐶))

Syntax hints:   = wceq 1542  Vcvv 3442   ∖ cdif 3900   ∩ cin 3902

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-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-in 3910

This theorem is referenced by:  ablfac1eulem  20018  ballotlemgun  34707