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Theorem nfeud 1964
Description: Deduction version of nfeu 1967. (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 1466 . . 3 𝑧𝜓
21sb8eu 1961 . 2 (∃!𝑦𝜓 ↔ ∃!𝑧[𝑧 / 𝑦]𝜓)
3 nfv 1466 . . 3 𝑧𝜑
4 nfeud.1 . . . 4 𝑦𝜑
5 nfeud.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfsbd 1899 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
73, 6nfeudv 1963 . 2 (𝜑 → Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜓)
82, 7nfxfrd 1409 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1394  [wsb 1692  ∃!weu 1948
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by:  nfmod  1965  hbeud  1970  nfreudxy  2540
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