Theorem inindir 4183

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 4174	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
2	1	ineq2i 4164	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶)
3		in4 4181	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))
4	2, 3	eqtr3i 2756	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1541   ∩ 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-in 3904

This theorem is referenced by:  difindir  4240  resindir  5944  predin  6274  restbas  23073  connsuba  23335  kgentopon  23453  trfbas2  23758  trfil2  23802  fclsrest  23939  trust  24144  chtdif  27095  ppidif  27100  mdslmd1lem1  32305  mdslmd1lem2  32306  mddmdin0i  32411  ballotlemgun  34538  cvmsss2  35318