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Theorem ssdifsym 3971
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 3970 . . . 4 ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
21ex 449 . . 3 (𝐴𝑉 → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
32adantr 472 . 2 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
4 ssdifim 3970 . . . 4 ((𝐵𝑉𝐴 = (𝑉𝐵)) → 𝐵 = (𝑉𝐴))
54ex 449 . . 3 (𝐵𝑉 → (𝐴 = (𝑉𝐵) → 𝐵 = (𝑉𝐴)))
65adantl 473 . 2 ((𝐴𝑉𝐵𝑉) → (𝐴 = (𝑉𝐵) → 𝐵 = (𝑉𝐴)))
73, 6impbid 202 1 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) ↔ 𝐴 = (𝑉𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  cdif 3677  wss 3680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rab 3023  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694
This theorem is referenced by: (None)
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