MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssdifsym Structured version   Visualization version   GIF version

Theorem ssdifsym 4274
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 4273 . . . 4 ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
21ex 412 . . 3 (𝐴𝑉 → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
32adantr 480 . 2 ((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) → 𝐴 = (𝑉𝐵)))
4 ssdifim 4273 . . . 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 1540  cdif 3948  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968
This theorem is referenced by:  zarcls  33873
  Copyright terms: Public domain W3C validator