Theorem inindir 4180

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 4171	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
2	1	ineq2i 4162	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶)
3		in4 4178	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))
4	2, 3	eqtr3i 2785	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3897

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 3905

This theorem is used by:  difindir  4238  resindir  5983  predin  6319  restbas  23437  connsuba  23699  kgentopon  23818  trfbas2  24123  trfil2  24167  fclsrest  24304  trust  24509  chtdif  27448  ppidif  27453  mdslmd1lem1  32860  mdslmd1lem2  32861  mddmdin0i  32966  ballotlemgun  35091  cvmsss2  35960