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Theorem eupicka 2079
 Description: Version of eupick 2078 with closed formulas. (Contributed by NM, 6-Sep-2008.)
Assertion
Ref Expression
eupicka ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))

Proof of Theorem eupicka
StepHypRef Expression
1 hbeu1 2009 . . 3 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
2 hbe1 1471 . . 3 (∃𝑥(𝜑𝜓) → ∀𝑥𝑥(𝜑𝜓))
31, 2hban 1526 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)))
4 eupick 2078 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
53, 4alrimih 1445 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329  ∃wex 1468  ∃!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-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003 This theorem is referenced by:  eupickbi  2081
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