Theorem bj-axdd2 34774

Description: This implication, proved using only ax-gen 1798 and ax-4 1812 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme ⊢ ∃𝑥⊤ implies the axiom scheme ⊢ (∀𝑥𝜑 → ∃𝑥𝜑). 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 34775. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-axdd2	⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))

Proof of Theorem bj-axdd2

Step	Hyp	Ref	Expression
1		ala1 1816	. . 3 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
2		exim 1836	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
3	1, 2	syl 17	. 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜓))
4	3	com12 32	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)