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Theorem eupicka 2028
Description: Version of eupick 2027 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 1958 . . 3 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
2 hbe1 1429 . . 3 (∃𝑥(𝜑𝜓) → ∀𝑥𝑥(𝜑𝜓))
31, 2hban 1484 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)))
4 eupick 2027 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
53, 4alrimih 1403 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287  wex 1426  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  eupickbi  2030
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