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Theorem bj-axdd2 31749
 Description: This implication, proved using only ax-gen 1713 and ax-4 1728 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 31750. (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
StepHypRef Expression
1 ala1 1755 . . 3 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
2 exim 1751 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
31, 2syl 17 . 2 (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜓))
43com12 32 1 (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by: (None)
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