Theorem bj-axdd2 37217

Description: This implication, proved using only ax-gen 1828 and ax-4 1842 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 37218. (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 1846	. . 3 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
2		exim 1867	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
3	1, 2	syl 18	. 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜓))
4	3	com12 33	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-spvw  37289  bj-spvew  37290