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

Theorem sbnv 1810
 Description: Version of sbn 1868 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 1808 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
2 alinexa 1535 . . 3 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
31, 2bitri 182 . 2 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
4 sb5 1809 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
53, 4xchbinxr 641 1 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1283  ∃wex 1422  [wsb 1686 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 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-sb 1687 This theorem is referenced by:  sbn  1868
 Copyright terms: Public domain W3C validator