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Theorem rabeq0 3241
Description: Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
Assertion
Ref Expression
rabeq0 ({x Aφ} = ∅ ↔ x A ¬ φ)

Proof of Theorem rabeq0
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 imnan 623 . . 3 ((x A → ¬ φ) ↔ ¬ (x A φ))
21albii 1356 . 2 (x(x A → ¬ φ) ↔ x ¬ (x A φ))
3 df-ral 2305 . 2 (x A ¬ φx(x A → ¬ φ))
4 sbn 1823 . . . 4 ([y / x] ¬ (x A φ) ↔ ¬ [y / x](x A φ))
54albii 1356 . . 3 (y[y / x] ¬ (x A φ) ↔ y ¬ [y / x](x A φ))
6 nfv 1418 . . . 4 y ¬ (x A φ)
76sb8 1733 . . 3 (x ¬ (x A φ) ↔ y[y / x] ¬ (x A φ))
8 eq0 3233 . . . 4 ({x Aφ} = ∅ ↔ y ¬ y {x Aφ})
9 df-rab 2309 . . . . . . . 8 {x Aφ} = {x ∣ (x A φ)}
109eleq2i 2101 . . . . . . 7 (y {x Aφ} ↔ y {x ∣ (x A φ)})
11 df-clab 2024 . . . . . . 7 (y {x ∣ (x A φ)} ↔ [y / x](x A φ))
1210, 11bitri 173 . . . . . 6 (y {x Aφ} ↔ [y / x](x A φ))
1312notbii 593 . . . . 5 y {x Aφ} ↔ ¬ [y / x](x A φ))
1413albii 1356 . . . 4 (y ¬ y {x Aφ} ↔ y ¬ [y / x](x A φ))
158, 14bitri 173 . . 3 ({x Aφ} = ∅ ↔ y ¬ [y / x](x A φ))
165, 7, 153bitr4ri 202 . 2 ({x Aφ} = ∅ ↔ x ¬ (x A φ))
172, 3, 163bitr4ri 202 1 ({x Aφ} = ∅ ↔ x A ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  [wsb 1642  {cab 2023  wral 2300  {crab 2304  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-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-nul 3219
This theorem is referenced by:  rabnc  3244  rabrsndc  3429  ssfiexmid  6254  iooidg  8508  icc0r  8525  fznlem  8635
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