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Theorem dif32 3343
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 3224 . . 3 (𝐵𝐶) = (𝐶𝐵)
21difeq2i 3195 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐶𝐵))
3 difun1 3340 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
4 difun1 3340 . 2 (𝐴 ∖ (𝐶𝐵)) = ((𝐴𝐶) ∖ 𝐵)
52, 3, 43eqtr3i 2169 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1332  cdif 3072  cun 3073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081
This theorem is referenced by:  difdifdirss  3451
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