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Theorem in13 3809
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 3808 . 2 ((𝐵𝐶) ∩ 𝐴) = ((𝐵𝐴) ∩ 𝐶)
2 incom 3788 . 2 (𝐴 ∩ (𝐵𝐶)) = ((𝐵𝐶) ∩ 𝐴)
3 incom 3788 . 2 (𝐶 ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ 𝐶)
41, 2, 33eqtr4i 2658 1 (𝐴 ∩ (𝐵𝐶)) = (𝐶 ∩ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cin 3559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-in 3567
This theorem is referenced by:  inin  29191
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