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Theorem in4 4217
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 4212 . . 3 (𝐵 ∩ (𝐶𝐷)) = (𝐶 ∩ (𝐵𝐷))
21ineq2i 4201 . 2 (𝐴 ∩ (𝐵 ∩ (𝐶𝐷))) = (𝐴 ∩ (𝐶 ∩ (𝐵𝐷)))
3 inass 4211 . 2 ((𝐴𝐵) ∩ (𝐶𝐷)) = (𝐴 ∩ (𝐵 ∩ (𝐶𝐷)))
4 inass 4211 . 2 ((𝐴𝐶) ∩ (𝐵𝐷)) = (𝐴 ∩ (𝐶 ∩ (𝐵𝐷)))
52, 3, 43eqtr4i 2762 1 ((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cin 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-in 3947
This theorem is referenced by:  inindi  4218  inindir  4219  fh2  31341  disjxpin  32288
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