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Theorem nfeu 1967
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 1466 . . 3 𝑧𝜑
21sb8eu 1961 . 2 (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑)
3 nfeu.1 . . . 4 𝑥𝜑
43nfsb 1870 . . 3 𝑥[𝑧 / 𝑦]𝜑
54nfeuv 1966 . 2 𝑥∃!𝑧[𝑧 / 𝑦]𝜑
62, 5nfxfr 1408 1 𝑥∃!𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wnf 1394  [wsb 1692  ∃!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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by:  hbeu  1969  eusv2nf  4278
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