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Theorem rabeq0 3224
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 611 . . 3 ((x A → ¬ φ) ↔ ¬ (x A φ))
21albii 1339 . 2 (x(x A → ¬ φ) ↔ x ¬ (x A φ))
3 df-ral 2289 . 2 (x A ¬ φx(x A → ¬ φ))
4 sbn 1808 . . . 4 ([y / x] ¬ (x A φ) ↔ ¬ [y / x](x A φ))
54albii 1339 . . 3 (y[y / x] ¬ (x A φ) ↔ y ¬ [y / x](x A φ))
6 nfv 1402 . . . 4 y ¬ (x A φ)
76sb8 1718 . . 3 (x ¬ (x A φ) ↔ y[y / x] ¬ (x A φ))
8 eq0 3216 . . . 4 ({x Aφ} = ∅ ↔ y ¬ y {x Aφ})
9 df-rab 2293 . . . . . . . 8 {x Aφ} = {x ∣ (x A φ)}
109eleq2i 2086 . . . . . . 7 (y {x Aφ} ↔ y {x ∣ (x A φ)})
11 df-clab 2009 . . . . . . 7 (y {x ∣ (x A φ)} ↔ [y / x](x A φ))
1210, 11bitri 173 . . . . . 6 (y {x Aφ} ↔ [y / x](x A φ))
1312notbii 581 . . . . 5 y {x Aφ} ↔ ¬ [y / x](x A φ))
1413albii 1339 . . . 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 1226   = wceq 1228   wcel 1374  [wsb 1627  {cab 2008  wral 2284  {crab 2288  c0 3201
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rab 2293  df-v 2537  df-dif 2897  df-nul 3202
This theorem is referenced by:  rabnc  3227  rabrsndc  3412
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