Theorem indifdir 4275

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 4274	. 2 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐵)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵))
2		incom 4189	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∖ 𝐵))
3		incom 4189	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		incom 4189	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
5	3, 4	difeq12i 4104	. 2 ⊢ ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵))
6	1, 2, 5	3eqtr4i 2769	1 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1540   ∖ cdif 3928   ∩ cin 3930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-in 3938

This theorem is referenced by:  resdifdir  6231  preddif  6323  fresaun  6754  uniioombllem4  25544  subsalsal  46355