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Theorem nfeu 2038
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 1521 . . 3 𝑧𝜑
21sb8eu 2032 . 2 (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑)
3 nfeu.1 . . . 4 𝑥𝜑
43nfsb 1939 . . 3 𝑥[𝑧 / 𝑦]𝜑
54nfeuv 2037 . 2 𝑥∃!𝑧[𝑧 / 𝑦]𝜑
62, 5nfxfr 1467 1 𝑥∃!𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wnf 1453  [wsb 1755  ∃!weu 2019
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022
This theorem is referenced by:  hbeu  2040  eusv2nf  4441
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