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Theorem hbeu 2020
Description: Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)
Hypothesis
Ref Expression
hbeu.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbeu (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

Proof of Theorem hbeu
StepHypRef Expression
1 hbeu.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21nfi 1438 . . 3 𝑥𝜑
32nfeu 2018 . 2 𝑥∃!𝑦𝜑
43nfri 1499 1 (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  ∃!weu 1999
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002
This theorem is referenced by:  hbmo  2038  2eu7  2093
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