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Theorem nfeuv 1915
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1916 but has the additional constraint that x and y must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)
Hypothesis
Ref Expression
nfeuv.1 xφ
Assertion
Ref Expression
nfeuv x∃!yφ
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem nfeuv
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfeuv.1 . . . . 5 xφ
2 nfv 1418 . . . . 5 x y = z
31, 2nfbi 1478 . . . 4 x(φy = z)
43nfal 1465 . . 3 xy(φy = z)
54nfex 1525 . 2 xzy(φy = z)
6 df-eu 1900 . . 3 (∃!yφzy(φy = z))
76nfbii 1359 . 2 (Ⅎx∃!yφ ↔ Ⅎxzy(φy = z))
85, 7mpbir 134 1 x∃!yφ
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240  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-tru 1245  df-nf 1347  df-eu 1900
This theorem is referenced by:  nfeu  1916
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