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Theorem nfeuv 2015
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2016 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 1508 . . . . 5 𝑥 𝑦 = 𝑧
31, 2nfbi 1568 . . . 4 𝑥(𝜑𝑦 = 𝑧)
43nfal 1555 . . 3 𝑥𝑦(𝜑𝑦 = 𝑧)
54nfex 1616 . 2 𝑥𝑧𝑦(𝜑𝑦 = 𝑧)
6 df-eu 2000 . . 3 (∃!𝑦𝜑 ↔ ∃𝑧𝑦(𝜑𝑦 = 𝑧))
76nfbii 1449 . 2 (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥𝑧𝑦(𝜑𝑦 = 𝑧))
85, 7mpbir 145 1 𝑥∃!𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1329  wnf 1436  wex 1468  ∃!weu 1997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-eu 2000
This theorem is referenced by:  nfeu  2016
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