Theorem indifdir 4314

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 4313	. 2 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐵)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵))
2		incom 4230	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∖ 𝐵))
3		incom 4230	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		incom 4230	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
5	3, 4	difeq12i 4147	. 2 ⊢ ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵))
6	1, 2, 5	3eqtr4i 2778	1 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1537   ∖ cdif 3973   ∩ cin 3975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983

This theorem is referenced by:  resdifdir  6268  preddif  6361  fresaun  6792  uniioombllem4  25640  subsalsal  46280