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Theorem in4 4221
Description: Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
Assertion
Ref Expression
in4 ((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))

Proof of Theorem in4
StepHypRef Expression
1 in12 4216 . . 3 (𝐵 ∩ (𝐶𝐷)) = (𝐶 ∩ (𝐵𝐷))
21ineq2i 4205 . 2 (𝐴 ∩ (𝐵 ∩ (𝐶𝐷))) = (𝐴 ∩ (𝐶 ∩ (𝐵𝐷)))
3 inass 4215 . 2 ((𝐴𝐵) ∩ (𝐶𝐷)) = (𝐴 ∩ (𝐵 ∩ (𝐶𝐷)))
4 inass 4215 . 2 ((𝐴𝐶) ∩ (𝐵𝐷)) = (𝐴 ∩ (𝐶 ∩ (𝐵𝐷)))
52, 3, 43eqtr4i 2765 1 ((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  cin 3943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-in 3951
This theorem is referenced by:  inindi  4222  inindir  4223  fh2  31422  disjxpin  32371
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