Theorem inindir 4236

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 4227	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
2	1	ineq2i 4217	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶)
3		in4 4234	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))
4	2, 3	eqtr3i 2767	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1540   ∩ cin 3950

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958

This theorem is referenced by:  difindir  4293  resindir  6014  predin  6348  restbas  23166  connsuba  23428  kgentopon  23546  trfbas2  23851  trfil2  23895  fclsrest  24032  trust  24238  chtdif  27201  ppidif  27206  mdslmd1lem1  32344  mdslmd1lem2  32345  mddmdin0i  32450  ballotlemgun  34527  cvmsss2  35279