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Theorem bj-axtd 32703
Description: This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme (∀𝑥𝜑𝜑) (modal T) implies the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑) (modal D). See also bj-axdd2 32701 and bj-axd2d 32702. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-axtd ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))

Proof of Theorem bj-axtd
StepHypRef Expression
1 con2 130 . . 3 ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → (𝜑 → ¬ ∀𝑥 ¬ 𝜑))
2 df-ex 1745 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
31, 2syl6ibr 242 . 2 ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → (𝜑 → ∃𝑥𝜑))
43imim2d 57 1 ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by: (None)
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