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Theorem eupicka 1996
Description: Version of eupick 1995 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 1926 . . 3 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
2 hbe1 1400 . . 3 (∃𝑥(𝜑𝜓) → ∀𝑥𝑥(𝜑𝜓))
31, 2hban 1455 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)))
4 eupick 1995 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
53, 4alrimih 1374 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920
This theorem is referenced by:  eupickbi  1998
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