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Theorem ssdifsym 4280
Description: Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.)
Assertion
Ref Expression
ssdifsym ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) ↔ 𝐴 = (𝑉𝐵)))

Proof of Theorem ssdifsym
StepHypRef Expression
1 ssdifim 4279 . . . 4 ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
21ex 412 . . 3 (𝐴𝑉 → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
32adantr 480 . 2 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
4 ssdifim 4279 . . . 4 ((𝐵𝑉𝐴 = (𝑉𝐵)) → 𝐵 = (𝑉𝐴))
54ex 412 . . 3 (𝐵𝑉 → (𝐴 = (𝑉𝐵) → 𝐵 = (𝑉𝐴)))
65adantl 481 . 2 ((𝐴𝑉𝐵𝑉) → (𝐴 = (𝑉𝐵) → 𝐵 = (𝑉𝐴)))
73, 6impbid 212 1 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) ↔ 𝐴 = (𝑉𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  cdif 3960  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980
This theorem is referenced by:  zarcls  33835
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