Theorem inindir 4186

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 4177	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
2	1	ineq2i 4167	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶)
3		in4 4184	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))
4	2, 3	eqtr3i 2759	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1541   ∩ cin 3898

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 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-in 3906

This theorem is referenced by:  difindir  4243  resindir  5953  predin  6283  restbas  23100  connsuba  23362  kgentopon  23480  trfbas2  23785  trfil2  23829  fclsrest  23966  trust  24171  chtdif  27122  ppidif  27127  mdslmd1lem1  32349  mdslmd1lem2  32350  mddmdin0i  32455  ballotlemgun  34631  cvmsss2  35417