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

Theorem nfeudv 1915
 Description: Deduction version of nfeu 1919. Similar to nfeud 1916 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)
Hypotheses
Ref Expression
nfeudv.1 𝑦𝜑
nfeudv.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfeudv (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfeudv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . 3 𝑧𝜑
2 nfeudv.1 . . . 4 𝑦𝜑
3 nfeudv.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
4 nfv 1421 . . . . . 6 𝑥 𝑦 = 𝑧
54a1i 9 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
63, 5nfbid 1480 . . . 4 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
72, 6nfald 1643 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
81, 7nfexd 1644 . 2 (𝜑 → Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
9 df-eu 1903 . . 3 (∃!𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
109nfbii 1362 . 2 (Ⅎ𝑥∃!𝑦𝜓 ↔ Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
118, 10sylibr 137 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  Ⅎwnf 1349  ∃wex 1381  ∃!weu 1900 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-eu 1903 This theorem is referenced by:  nfeud  1916
 Copyright terms: Public domain W3C validator