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Theorem hbeu1 1926
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 1919 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 hba1 1449 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑥(𝜑𝑥 = 𝑦))
32hbex 1543 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦𝑥(𝜑𝑥 = 𝑦))
41, 3hbxfrbi 1377 1 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wex 1397  ∃!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-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-eu 1919
This theorem is referenced by:  hbmo1  1954  eupicka  1996  exists2  2013
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