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

Theorem sbnv 1816
Description: Version of sbn 1874 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
Assertion
Ref Expression
sbnv ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbnv
StepHypRef Expression
1 sb6 1814 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
2 alinexa 1539 . . 3 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
31, 2bitri 182 . 2 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
4 sb5 1815 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
53, 4xchbinxr 643 1 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wal 1287  wex 1426  [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-sb 1693
This theorem is referenced by:  sbn  1874
  Copyright terms: Public domain W3C validator