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Theorem abeq0 3242
 Description: Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
Assertion
Ref Expression
abeq0 ({xφ} = ∅ ↔ x ¬ φ)

Proof of Theorem abeq0
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sbn 1823 . . 3 ([y / x] ¬ φ ↔ ¬ [y / x]φ)
21albii 1356 . 2 (y[y / x] ¬ φy ¬ [y / x]φ)
3 nfv 1418 . . 3 y ¬ φ
43sb8 1733 . 2 (x ¬ φy[y / x] ¬ φ)
5 eq0 3233 . . 3 ({xφ} = ∅ ↔ y ¬ y {xφ})
6 df-clab 2024 . . . . 5 (y {xφ} ↔ [y / x]φ)
76notbii 593 . . . 4 y {xφ} ↔ ¬ [y / x]φ)
87albii 1356 . . 3 (y ¬ y {xφ} ↔ y ¬ [y / x]φ)
95, 8bitri 173 . 2 ({xφ} = ∅ ↔ y ¬ [y / x]φ)
102, 4, 93bitr4ri 202 1 ({xφ} = ∅ ↔ x ¬ φ)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  [wsb 1642  {cab 2023  ∅c0 3218 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219 This theorem is referenced by:  opprc  3561
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