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Theorem dif32 4034
 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 3900 . . 3 (𝐵𝐶) = (𝐶𝐵)
21difeq2i 3868 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐶𝐵))
3 difun1 4030 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
4 difun1 4030 . 2 (𝐴 ∖ (𝐶𝐵)) = ((𝐴𝐶) ∖ 𝐵)
52, 3, 43eqtr3i 2790 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∖ cdif 3712   ∪ cun 3713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722 This theorem is referenced by:  difdifdir  4200  difsnen  8207  nbupgruvtxres  26512  poimirlem25  33747
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