Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  euim GIF version

Theorem euim 1968
 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 5 . . 3 (∃𝑥𝜑 → (∃!𝑥𝜓 → ∃𝑥𝜑))
2 euimmo 1967 . . 3 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
31, 2anim12ii 325 . 2 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑)))
4 eu5 1947 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
53, 4syl6ibr 151 1 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381  ∃!weu 1900  ∃*wmo 1901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator