Theorem indifdir 4242

Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by BTernaryTau, 14-Aug-2024.)

Assertion

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

Proof of Theorem indifdir

Step	Hyp	Ref	Expression
1		indifdi 4241	. 2 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐵)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵))
2		incom 4156	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∖ 𝐵))
3		incom 4156	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		incom 4156	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
5	3, 4	difeq12i 4071	. 2 ⊢ ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵))
6	1, 2, 5	3eqtr4i 2764	1 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1541   ∖ cdif 3894   ∩ cin 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-in 3904

This theorem is referenced by:  resdifdir  6184  preddif  6276  fresaun  6694  uniioombllem4  25514  subsalsal  46467