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Theorem nfeuv 1918
 Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1919 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 1421 . . . . 5 𝑥 𝑦 = 𝑧
31, 2nfbi 1481 . . . 4 𝑥(𝜑𝑦 = 𝑧)
43nfal 1468 . . 3 𝑥𝑦(𝜑𝑦 = 𝑧)
54nfex 1528 . 2 𝑥𝑧𝑦(𝜑𝑦 = 𝑧)
6 df-eu 1903 . . 3 (∃!𝑦𝜑 ↔ ∃𝑧𝑦(𝜑𝑦 = 𝑧))
76nfbii 1362 . 2 (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥𝑧𝑦(𝜑𝑦 = 𝑧))
85, 7mpbir 134 1 𝑥∃!𝑦𝜑
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  ∃!weu 1900 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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-eu 1903 This theorem is referenced by:  nfeu  1919
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