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Theorem difcom 4433
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 4100 . . 3 (𝐵𝐶) = (𝐶𝐵)
21sseq2i 3961 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴 ⊆ (𝐶𝐵))
3 ssundif 4432 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
4 ssundif 4432 . 2 (𝐴 ⊆ (𝐶𝐵) ↔ (𝐴𝐶) ⊆ 𝐵)
52, 3, 43bitr3i 300 1 ((𝐴𝐵) ⊆ 𝐶 ↔ (𝐴𝐶) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  cdif 3895  cun 3896  wss 3898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915
This theorem is referenced by:  pssdifcom1  4434  pssdifcom2  4435  isreg2  22634  restmetu  23832  conss1  42391  icccncfext  43772
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