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Theorem nfeu 1935
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 1437 . . 3 𝑧𝜑
21sb8eu 1929 . 2 (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑)
3 nfeu.1 . . . 4 𝑥𝜑
43nfsb 1838 . . 3 𝑥[𝑧 / 𝑦]𝜑
54nfeuv 1934 . 2 𝑥∃!𝑧[𝑧 / 𝑦]𝜑
62, 5nfxfr 1379 1 𝑥∃!𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wnf 1365  [wsb 1661  ∃!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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919
This theorem is referenced by:  hbeu  1937  eusv2nf  4215
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