New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  euim GIF version

Theorem euim 2254
 Description: Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
euim ((xφ x(φψ)) → (∃!xψ∃!xφ))

Proof of Theorem euim
StepHypRef Expression
1 ax-1 6 . . 3 (xφ → (∃!xψxφ))
2 euimmo 2253 . . 3 (x(φψ) → (∃!xψ∃*xφ))
31, 2anim12ii 553 . 2 ((xφ x(φψ)) → (∃!xψ → (xφ ∃*xφ)))
4 eu5 2242 . 2 (∃!xφ ↔ (xφ ∃*xφ))
53, 4syl6ibr 218 1 ((xφ x(φψ)) → (∃!xψ∃!xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator