Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  abeq0 GIF version

Theorem abeq0 3393
 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 1925 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
21albii 1446 . 2 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
3 nfv 1508 . . 3 𝑦 ¬ 𝜑
43sb8 1828 . 2 (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)
5 eq0 3381 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
6 df-clab 2126 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
76notbii 657 . . . 4 𝑦 ∈ {𝑥𝜑} ↔ ¬ [𝑦 / 𝑥]𝜑)
87albii 1446 . . 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 1329   = wceq 1331   ∈ wcel 1480  [wsb 1735  {cab 2125  ∅c0 3363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364 This theorem is referenced by:  opprc  3726
 Copyright terms: Public domain W3C validator