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Theorem difcom 4481
Description: Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
Assertion
Ref Expression
difcom ((𝐴𝐵) ⊆ 𝐶 ↔ (𝐴𝐶) ⊆ 𝐵)

Proof of Theorem difcom
StepHypRef Expression
1 uncom 4146 . . 3 (𝐵𝐶) = (𝐶𝐵)
21sseq2i 4004 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴 ⊆ (𝐶𝐵))
3 ssundif 4480 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
4 ssundif 4480 . 2 (𝐴 ⊆ (𝐶𝐵) ↔ (𝐴𝐶) ⊆ 𝐵)
52, 3, 43bitr3i 301 1 ((𝐴𝐵) ⊆ 𝐶 ↔ (𝐴𝐶) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  cdif 3938  cun 3939  wss 3941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958
This theorem is referenced by:  pssdifcom1  4482  pssdifcom2  4483  isreg2  23225  restmetu  24423  conss1  43753  icccncfext  45149
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