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

Theorem nfeud 2013
 Description: Deduction version of nfeu 2016. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
Hypotheses
Ref Expression
nfeud.1 𝑦𝜑
nfeud.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfeud (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1508 . . 3 𝑧𝜓
21sb8eu 2010 . 2 (∃!𝑦𝜓 ↔ ∃!𝑧[𝑧 / 𝑦]𝜓)
3 nfv 1508 . . 3 𝑧𝜑
4 nfeud.1 . . . 4 𝑦𝜑
5 nfeud.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfsbd 1948 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
73, 6nfeudv 2012 . 2 (𝜑 → Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜓)
82, 7nfxfrd 1451 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1436  [wsb 1735  ∃!weu 1997 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-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 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000 This theorem is referenced by:  nfmod  2014  hbeud  2019  nfreudxy  2602
 Copyright terms: Public domain W3C validator