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

Theorem eupickb 2699
Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
Assertion
Ref Expression
eupickb ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem eupickb
StepHypRef Expression
1 eupick 2697 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
213adant2 1154 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
3 exancom 1947 . . . 4 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 eupick 2697 . . . 4 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓𝜑)) → (𝜓𝜑))
53, 4sylan2b 583 . . 3 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
653adant1 1153 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
72, 6impbid 203 1 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100  wex 1859  ∃!weu 2629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-10 2184  ax-12 2213
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-ex 1860  df-nf 1864  df-eu 2633  df-mo 2634
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator