Theorem bj-axtd 31751

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 31749 and bj-axd2d 31750. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-axtd	⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑 → 𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))

Proof of Theorem bj-axtd

Step	Hyp	Ref	Expression
1		con2 129	. . 3 ⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → (𝜑 → ¬ ∀𝑥 ¬ 𝜑))
2		df-ex 1696	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	syl6ibr 241	. 2 ⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → (𝜑 → ∃𝑥𝜑))
4	3	imim2d 55	1 ⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑 → 𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695

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 196  df-ex 1696

This theorem is referenced by: (None)