Theorem in12 4151

Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)

Assertion

Ref	Expression
in12	⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶))

Proof of Theorem in12

Step	Hyp	Ref	Expression
1		incom 4131	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2	1	ineq1i 4139	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
3		inass 4150	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
4		inass 4150	. 2 ⊢ ((𝐵 ∩ 𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴 ∩ 𝐶))
5	2, 3, 4	3eqtr3i 2774	1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶))

Syntax hints:   = wceq 1539   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890

This theorem is referenced by:  in32  4152  in31  4154  in4  4156  resdmres  6124  kmlem12  9848  ressress  16884  fh1  29881  fh2  29882  mdslmd3i  30595  bj-inrab3  35044