Theorem inindi 3448

Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994.)

Assertion

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

Proof of Theorem inindi

Step	Hyp	Ref	Expression
1		inidm 3440	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
2	1	ineq1i 3428	. 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ (𝐵 ∩ 𝐶))
3		in4 3447	. 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶))
4	2, 3	eqtr3i 2261	1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶))

Syntax hints:   = wceq 1402   ∩ cin 3219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is referenced by:  resindi  5073  offres  6358  bitsinv1  12707