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Theorem ab0 3569
 Description: Empty class abstraction. (Contributed by SF, 5-Jan-2018.)
Assertion
Ref Expression
ab0 ({x φ} = x ¬ φ)

Proof of Theorem ab0
StepHypRef Expression
1 abn0 3568 . . 3 ({x φ} ≠ xφ)
2 df-ne 2518 . . 3 ({x φ} ≠ ↔ ¬ {x φ} = )
3 df-ex 1542 . . 3 (xφ ↔ ¬ x ¬ φ)
41, 2, 33bitr3i 266 . 2 (¬ {x φ} = ↔ ¬ x ¬ φ)
54con4bii 288 1 ({x φ} = x ¬ φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  {cab 2339   ≠ wne 2516  ∅c0 3550 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-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by:  setswith  4321  nnadjoin  4520  tfinnn  4534
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