Theorem bj-axd2d

Description: This implication, proved using only ax-gen on top of propositional calculus (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme |- ( A. x ph -> E. x ph ) implies the axiom scheme |- E. x T. . 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 . (Contributed by BJ, 16-May-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-axd2d	⊢ ( ( ∀ 𝑥 ⊤ → ∃ 𝑥 ⊤ ) → ∃ 𝑥 ⊤ )

Proof

Step	Hyp	Ref	Expression
1		pm2.27	⊢ ( ∀ 𝑥 ⊤ → ( ( ∀ 𝑥 ⊤ → ∃ 𝑥 ⊤ ) → ∃ 𝑥 ⊤ ) )
2		tru	⊢ ⊤
3	1 2	mpg	⊢ ( ( ∀ 𝑥 ⊤ → ∃ 𝑥 ⊤ ) → ∃ 𝑥 ⊤ )