Theorem bj-gl4

Description: In a normal modal logic, the modal axiom GL implies the modal axiom (4). Translated to first-order logic, Axiom GL reads |- ( A. x ( A. x ph -> ph ) -> A. x ph ) . Note that the antecedent of bj-gl4 is an instance of the axiom GL, with ph replaced by ( A. x ph /\ ph ) , which is a modality sometimes called the "strong necessity" of ph . (Contributed by BJ, 12-Dec-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-gl4	⊢ ( ( ∀ 𝑥 ( ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) → ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) → ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		19.26	⊢ ( ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ↔ ( ∀ 𝑥 ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜑 ) )
2		simpr	⊢ ( ( ∀ 𝑥 ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜑 ) → ∀ 𝑥 𝜑 )
3	2	a1i	⊢ ( 𝜑 → ( ( ∀ 𝑥 ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜑 ) → ∀ 𝑥 𝜑 ) )
4	3	anc2ri	⊢ ( 𝜑 → ( ( ∀ 𝑥 ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜑 ) → ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) )
5	1 4	biimtrid	⊢ ( 𝜑 → ( ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) → ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) )
6	5	alimi	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) → ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) )
7	1	biimpi	⊢ ( ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) → ( ∀ 𝑥 ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜑 ) )
8	6 7	imim12i	⊢ ( ( ∀ 𝑥 ( ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) → ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) → ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) → ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜑 ) ) )
9		simpl	⊢ ( ( ∀ 𝑥 ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜑 ) → ∀ 𝑥 ∀ 𝑥 𝜑 )
10	8 9	syl6	⊢ ( ( ∀ 𝑥 ( ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) → ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) → ∀ 𝑥 ( ∀ 𝑥 𝜑 ∧ 𝜑 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) )