MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eupicka Structured version   Visualization version   GIF version

Theorem eupicka 2566
Description: Version of eupick 2565 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 nfeu1 2508 . . 3 𝑥∃!𝑥𝜑
2 nfe1 2067 . . 3 𝑥𝑥(𝜑𝜓)
31, 2nfan 1868 . 2 𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))
4 eupick 2565 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
53, 4alrimi 2120 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wex 1744  ∃!weu 2498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503
This theorem is referenced by:  eupickbi  2568  frege124d  38370  sbiota1  38952
  Copyright terms: Public domain W3C validator