Theorem inindi 4194

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 4186	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
2	1	ineq1i 4175	. 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ (𝐵 ∩ 𝐶))
3		in4 4193	. 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶))
4	2, 3	eqtr3i 2754	1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶))

Syntax hints:   = wceq 1540   ∩ cin 3910

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-in 3918

This theorem is referenced by:  difundi  4249  dfif5  4501  resindi  5955  offres  7941  incexclem  15778  bitsinv1  16388  bitsinvp1  16395  bitsres  16419  fh1  31597