Theorem bj-axd2d 36818

Description: This implication, proved using only ax-gen 1797 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⊤ (substitute ⊤ for 𝜑). 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 36817. (Contributed by BJ, 16-May-2019.) Generalize from its instance with ⊤ substituted for 𝜑. (Revised by BJ, 20-Mar-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem bj-axd2d

Step	Hyp	Ref	Expression
1		pm2.27 42	. 2 ⊢ (∀𝑥⊤ → ((∀𝑥⊤ → ∃𝑥𝜑) → ∃𝑥𝜑))
2		tru 1546	. 2 ⊢ ⊤
3	1, 2	mpg 1799	1 ⊢ ((∀𝑥⊤ → ∃𝑥𝜑) → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1540  ⊤wtru 1543  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797

This theorem depends on definitions:  df-bi 207  df-tru 1545

This theorem is referenced by: (None)