Theorem bj-axdd2

Description: This implication, proved using only ax-gen and ax-4 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme |- E. x T. implies the axiom scheme |- ( A. x ph -> E. x ph ) . 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-axd2d . (Contributed by BJ, 16-May-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-axdd2	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ala1	⊢ ( ∀ 𝑥 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
2		exim	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3	1 2	syl	⊢ ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
4	3	com12	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜓 ) )