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Theorem ssdifsym 4223
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 4222 . . . 4 ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
21ex 416 . . 3 (𝐴𝑉 → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
32adantr 484 . 2 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
4 ssdifim 4222 . . . 4 ((𝐵𝑉𝐴 = (𝑉𝐵)) → 𝐵 = (𝑉𝐴))
54ex 416 . . 3 (𝐵𝑉 → (𝐴 = (𝑉𝐵) → 𝐵 = (𝑉𝐴)))
65adantl 485 . 2 ((𝐴𝑉𝐵𝑉) → (𝐴 = (𝑉𝐵) → 𝐵 = (𝑉𝐴)))
73, 6impbid 215 1 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) ↔ 𝐴 = (𝑉𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  cdif 3915  wss 3918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3141  df-v 3481  df-dif 3921  df-in 3925  df-ss 3935
This theorem is referenced by: (None)
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