NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  complab GIF version

Theorem complab 3524
Description: Complement of a class abstraction. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
complab ∼ {x φ} = {x ¬ φ}

Proof of Theorem complab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-clab 2340 . . . . 5 (y {x φ} ↔ [y / x]φ)
21notbii 287 . . . 4 y {x φ} ↔ ¬ [y / x]φ)
3 sbn 2062 . . . 4 ([y / x] ¬ φ ↔ ¬ [y / x]φ)
42, 3bitr4i 243 . . 3 y {x φ} ↔ [y / x] ¬ φ)
5 vex 2862 . . . 4 y V
65elcompl 3225 . . 3 (y ∼ {x φ} ↔ ¬ y {x φ})
7 df-clab 2340 . . 3 (y {x ¬ φ} ↔ [y / x] ¬ φ)
84, 6, 73bitr4i 268 . 2 (y ∼ {x φ} ↔ y {x ¬ φ})
98eqriv 2350 1 ∼ {x φ} = {x ¬ φ}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1642  [wsb 1648   wcel 1710  {cab 2339  ccompl 3205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212
This theorem is referenced by:  nulnnn  4556  addccan2nclem2  6264
  Copyright terms: Public domain W3C validator