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Theorem in4 3202
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 3197 . . 3 (𝐵 ∩ (𝐶𝐷)) = (𝐶 ∩ (𝐵𝐷))
21ineq2i 3184 . 2 (𝐴 ∩ (𝐵 ∩ (𝐶𝐷))) = (𝐴 ∩ (𝐶 ∩ (𝐵𝐷)))
3 inass 3196 . 2 ((𝐴𝐵) ∩ (𝐶𝐷)) = (𝐴 ∩ (𝐵 ∩ (𝐶𝐷)))
4 inass 3196 . 2 ((𝐴𝐶) ∩ (𝐵𝐷)) = (𝐴 ∩ (𝐶 ∩ (𝐵𝐷)))
52, 3, 43eqtr4i 2115 1 ((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:   = wceq 1287  cin 2985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-in 2992
This theorem is referenced by:  inindi  3203  inindir  3204
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