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Theorem in4 3217
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 3212 . . 3 (𝐵 ∩ (𝐶𝐷)) = (𝐶 ∩ (𝐵𝐷))
21ineq2i 3199 . 2 (𝐴 ∩ (𝐵 ∩ (𝐶𝐷))) = (𝐴 ∩ (𝐶 ∩ (𝐵𝐷)))
3 inass 3211 . 2 ((𝐴𝐵) ∩ (𝐶𝐷)) = (𝐴 ∩ (𝐵 ∩ (𝐶𝐷)))
4 inass 3211 . 2 ((𝐴𝐶) ∩ (𝐵𝐷)) = (𝐴 ∩ (𝐶 ∩ (𝐵𝐷)))
52, 3, 43eqtr4i 2119 1 ((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:   = wceq 1290  cin 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006
This theorem is referenced by:  inindi  3218  inindir  3219
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