Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-axd2d Structured version   Visualization version   GIF version

Theorem bj-axd2d 32552
Description: This implication, proved using only ax-gen 1720 on top of propositional calculus (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑) implies the axiom scheme 𝑥. These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axdd2 32551. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-axd2d ((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)

Proof of Theorem bj-axd2d
StepHypRef Expression
1 pm2.27 42 . 2 (∀𝑥⊤ → ((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤))
2 tru 1485 . 2
31, 2mpg 1722 1 ((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479  wtru 1482  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720
This theorem depends on definitions:  df-bi 197  df-tru 1484
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator