Theorem inindir 4216

Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
inindir	⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))

Proof of Theorem inindir

Step	Hyp	Ref	Expression
1		inidm 4207	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
2	1	ineq2i 4197	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶)
3		in4 4214	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))
4	2, 3	eqtr3i 2761	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1540   ∩ 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-in 3938

This theorem is referenced by:  difindir  4273  resindir  5988  predin  6321  restbas  23101  connsuba  23363  kgentopon  23481  trfbas2  23786  trfil2  23830  fclsrest  23967  trust  24173  chtdif  27125  ppidif  27130  mdslmd1lem1  32311  mdslmd1lem2  32312  mddmdin0i  32417  ballotlemgun  34562  cvmsss2  35301