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Theorem abeq0 3468
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 1964 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
21albii 1481 . 2 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
3 nfv 1539 . . 3 𝑦 ¬ 𝜑
43sb8 1867 . 2 (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)
5 eq0 3456 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
6 df-clab 2176 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
76notbii 669 . . . 4 𝑦 ∈ {𝑥𝜑} ↔ ¬ [𝑦 / 𝑥]𝜑)
87albii 1481 . . 3 (∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
95, 8bitri 184 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
102, 4, 93bitr4ri 213 1 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105  wal 1362   = wceq 1364  [wsb 1773  wcel 2160  {cab 2175  c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438
This theorem is referenced by:  opprc  3814
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