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Theorem hbeu1 1910
 Description: Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
Assertion
Ref Expression
hbeu1 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)

Proof of Theorem hbeu1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-eu 1903 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 hba1 1433 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑥(𝜑𝑥 = 𝑦))
32hbex 1527 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦𝑥(𝜑𝑥 = 𝑦))
41, 3hbxfrbi 1361 1 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381  ∃!weu 1900 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-eu 1903 This theorem is referenced by:  hbmo1  1938  eupicka  1980  exists2  1997
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