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Theorem abeq0 3317
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 1875 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
21albii 1405 . 2 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
3 nfv 1467 . . 3 𝑦 ¬ 𝜑
43sb8 1785 . 2 (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)
5 eq0 3305 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
6 df-clab 2076 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
76notbii 630 . . . 4 𝑦 ∈ {𝑥𝜑} ↔ ¬ [𝑦 / 𝑥]𝜑)
87albii 1405 . . 3 (∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
95, 8bitri 183 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
102, 4, 93bitr4ri 212 1 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wal 1288   = wceq 1290  wcel 1439  [wsb 1693  {cab 2075  c0 3287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-nul 3288
This theorem is referenced by:  opprc  3649
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