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Theorem nfeuv 2063
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2064 but has the additional condition 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 1542 . . . . 5 𝑥 𝑦 = 𝑧
31, 2nfbi 1603 . . . 4 𝑥(𝜑𝑦 = 𝑧)
43nfal 1590 . . 3 𝑥𝑦(𝜑𝑦 = 𝑧)
54nfex 1651 . 2 𝑥𝑧𝑦(𝜑𝑦 = 𝑧)
6 df-eu 2048 . . 3 (∃!𝑦𝜑 ↔ ∃𝑧𝑦(𝜑𝑦 = 𝑧))
76nfbii 1487 . 2 (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥𝑧𝑦(𝜑𝑦 = 𝑧))
85, 7mpbir 146 1 𝑥∃!𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wb 105  wal 1362  wnf 1474  wex 1506  ∃!weu 2045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-i5r 1549
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-eu 2048
This theorem is referenced by:  nfeu  2064
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