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Theorem nfeudv 1912
Description: Deduction version of nfeu 1916. Similar to nfeud 1913 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 1418 . . 3 zφ
2 nfeudv.1 . . . 4 yφ
3 nfeudv.2 . . . . 5 (φ → Ⅎxψ)
4 nfv 1418 . . . . . 6 x y = z
54a1i 9 . . . . 5 (φ → Ⅎx y = z)
63, 5nfbid 1477 . . . 4 (φ → Ⅎx(ψy = z))
72, 6nfald 1640 . . 3 (φ → Ⅎxy(ψy = z))
81, 7nfexd 1641 . 2 (φ → Ⅎxzy(ψy = z))
9 df-eu 1900 . . 3 (∃!yψzy(ψy = z))
109nfbii 1359 . 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 1240   = wceq 1242  wnf 1346  wex 1378  ∃!weu 1897
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-eu 1900
This theorem is referenced by:  nfeud  1913
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