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Theorem dif32 3398
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 3279 . . 3 (𝐵𝐶) = (𝐶𝐵)
21difeq2i 3250 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐶𝐵))
3 difun1 3395 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
4 difun1 3395 . 2 (𝐴 ∖ (𝐶𝐵)) = ((𝐴𝐶) ∖ 𝐵)
52, 3, 43eqtr3i 2206 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  cdif 3126  cun 3127
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135
This theorem is referenced by:  difdifdirss  3507
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