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Theorem nfeuv 1966
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1967 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 1466 . . . . 5 𝑥 𝑦 = 𝑧
31, 2nfbi 1526 . . . 4 𝑥(𝜑𝑦 = 𝑧)
43nfal 1513 . . 3 𝑥𝑦(𝜑𝑦 = 𝑧)
54nfex 1573 . 2 𝑥𝑧𝑦(𝜑𝑦 = 𝑧)
6 df-eu 1951 . . 3 (∃!𝑦𝜑 ↔ ∃𝑧𝑦(𝜑𝑦 = 𝑧))
76nfbii 1407 . 2 (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥𝑧𝑦(𝜑𝑦 = 𝑧))
85, 7mpbir 144 1 𝑥∃!𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1287  wnf 1394  wex 1426  ∃!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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-eu 1951
This theorem is referenced by:  nfeu  1967
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