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Theorem nfeuv 1934
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1935 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 1437 . . . . 5 𝑥 𝑦 = 𝑧
31, 2nfbi 1497 . . . 4 𝑥(𝜑𝑦 = 𝑧)
43nfal 1484 . . 3 𝑥𝑦(𝜑𝑦 = 𝑧)
54nfex 1544 . 2 𝑥𝑧𝑦(𝜑𝑦 = 𝑧)
6 df-eu 1919 . . 3 (∃!𝑦𝜑 ↔ ∃𝑧𝑦(𝜑𝑦 = 𝑧))
76nfbii 1378 . 2 (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥𝑧𝑦(𝜑𝑦 = 𝑧))
85, 7mpbir 138 1 𝑥∃!𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257  wnf 1365  wex 1397  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-eu 1919
This theorem is referenced by:  nfeu  1935
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