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Theorem ssdifsym 4090
 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 4089 . . . 4 ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
21ex 403 . . 3 (𝐴𝑉 → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
32adantr 474 . 2 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
4 ssdifim 4089 . . . 4 ((𝐵𝑉𝐴 = (𝑉𝐵)) → 𝐵 = (𝑉𝐴))
54ex 403 . . 3 (𝐵𝑉 → (𝐴 = (𝑉𝐵) → 𝐵 = (𝑉𝐴)))
65adantl 475 . 2 ((𝐴𝑉𝐵𝑉) → (𝐴 = (𝑉𝐵) → 𝐵 = (𝑉𝐴)))
73, 6impbid 204 1 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) ↔ 𝐴 = (𝑉𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∖ cdif 3789   ⊆ wss 3792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-in 3799  df-ss 3806 This theorem is referenced by: (None)
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