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Theorem bj-axtd 33157
 Description: This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme ⊢ (∀𝑥𝜑 → 𝜑) (modal T) implies the axiom scheme ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) (modal D). See also bj-axdd2 33155 and bj-axd2d 33156. (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 133 . . 3 ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → (𝜑 → ¬ ∀𝑥 ¬ 𝜑))
2 df-ex 1824 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
31, 2syl6ibr 244 . 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 1599  ∃wex 1823 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 199  df-ex 1824 This theorem is referenced by: (None)
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