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Theorem nfeud 1958
Description: Deduction version of nfeu 1961. (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 1462 . . 3 𝑧𝜓
21sb8eu 1955 . 2 (∃!𝑦𝜓 ↔ ∃!𝑧[𝑧 / 𝑦]𝜓)
3 nfv 1462 . . 3 𝑧𝜑
4 nfeud.1 . . . 4 𝑦𝜑
5 nfeud.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfsbd 1893 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
73, 6nfeudv 1957 . 2 (𝜑 → Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜓)
82, 7nfxfrd 1405 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1390  [wsb 1686  ∃!weu 1942
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945
This theorem is referenced by:  nfmod  1959  hbeud  1964  nfreudxy  2528
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