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Theorem dif32 3177
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 3064 . . 3 (B𝐶) = (𝐶B)
21difeq2i 3036 . 2 (A ∖ (B𝐶)) = (A ∖ (𝐶B))
3 difun1 3174 . 2 (A ∖ (B𝐶)) = ((AB) ∖ 𝐶)
4 difun1 3174 . 2 (A ∖ (𝐶B)) = ((A𝐶) ∖ B)
52, 3, 43eqtr3i 2050 1 ((AB) ∖ 𝐶) = ((A𝐶) ∖ B)
Colors of variables: wff set class
Syntax hints:   = wceq 1228  cdif 2891  cun 2892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901
This theorem is referenced by:  difdifdirss  3284
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