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Theorem in31 3151
 Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
Assertion
Ref Expression
in31 ((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)

Proof of Theorem in31
StepHypRef Expression
1 in12 3148 . 2 (𝐶 ∩ (𝐴𝐵)) = (𝐴 ∩ (𝐶𝐵))
2 incom 3129 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴𝐵))
3 incom 3129 . 2 ((𝐶𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐶𝐵))
41, 2, 33eqtr4i 2070 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∩ cin 2916 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924 This theorem is referenced by:  inrot  3152
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