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Theorem nfeu 2025
Description: Bound-variable hypothesis builder for existential uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.)
Hypothesis
Ref Expression
nfeu.1 𝑥𝜑
Assertion
Ref Expression
nfeu 𝑥∃!𝑦𝜑

Proof of Theorem nfeu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1508 . . 3 𝑧𝜑
21sb8eu 2019 . 2 (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑)
3 nfeu.1 . . . 4 𝑥𝜑
43nfsb 1926 . . 3 𝑥[𝑧 / 𝑦]𝜑
54nfeuv 2024 . 2 𝑥∃!𝑧[𝑧 / 𝑦]𝜑
62, 5nfxfr 1454 1 𝑥∃!𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wnf 1440  [wsb 1742  ∃!weu 2006
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009
This theorem is referenced by:  hbeu  2027  eusv2nf  4417
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