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Theorem in13 3177
Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
Assertion
Ref Expression
in13 (𝐴 ∩ (𝐵𝐶)) = (𝐶 ∩ (𝐵𝐴))

Proof of Theorem in13
StepHypRef Expression
1 in32 3176 . 2 ((𝐵𝐶) ∩ 𝐴) = ((𝐵𝐴) ∩ 𝐶)
2 incom 3156 . 2 (𝐴 ∩ (𝐵𝐶)) = ((𝐵𝐶) ∩ 𝐴)
3 incom 3156 . 2 (𝐶 ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ 𝐶)
41, 2, 33eqtr4i 2086 1 (𝐴 ∩ (𝐵𝐶)) = (𝐶 ∩ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cin 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951
This theorem is referenced by: (None)
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