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Theorem bj-axd2d 33920
Description: This implication, proved using only ax-gen 1790 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 33919. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
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 1535 . 2
31, 2mpg 1792 1 ((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1529  wtru 1532  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790
This theorem depends on definitions:  df-bi 209  df-tru 1534
This theorem is referenced by: (None)
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