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

Theorem exmodc 1993
 Description: If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
Assertion
Ref Expression
exmodc (DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))

Proof of Theorem exmodc
StepHypRef Expression
1 df-dc 777 . 2 (DECID𝑥𝜑 ↔ (∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2 pm2.21 580 . . . 4 (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
3 df-mo 1947 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
42, 3sylibr 132 . . 3 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
54orim2i 711 . 2 ((∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑) → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
61, 5sylbi 119 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 662  DECID wdc 776  ∃wex 1422  ∃!weu 1943  ∃*wmo 1944 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663 This theorem depends on definitions:  df-bi 115  df-dc 777  df-mo 1947 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator