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Theorem dif32 3418
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 3299 . . 3 (𝐵𝐶) = (𝐶𝐵)
21difeq2i 3270 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐶𝐵))
3 difun1 3415 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
4 difun1 3415 . 2 (𝐴 ∖ (𝐶𝐵)) = ((𝐴𝐶) ∖ 𝐵)
52, 3, 43eqtr3i 2218 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1364  cdif 3146  cun 3147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2758  df-dif 3151  df-un 3153  df-in 3155
This theorem is referenced by:  difdifdirss  3527
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