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Theorem dif32 3243
Description: Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
Assertion
Ref Expression
dif32 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)

Proof of Theorem dif32
StepHypRef Expression
1 uncom 3126 . . 3 (𝐵𝐶) = (𝐶𝐵)
21difeq2i 3097 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐶𝐵))
3 difun1 3240 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
4 difun1 3240 . 2 (𝐴 ∖ (𝐶𝐵)) = ((𝐴𝐶) ∖ 𝐵)
52, 3, 43eqtr3i 2111 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1285  cdif 2979  cun 2980
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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988
This theorem is referenced by:  difdifdirss  3343
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