Theorem bj-gl4 34514

Description: In a normal modal logic, the modal axiom GL implies the modal axiom (4). Translated to first-order logic, Axiom GL reads ⊢ (∀𝑥(∀𝑥𝜑 → 𝜑) → ∀𝑥𝜑). Note that the antecedent of bj-gl4 34514 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑 ∧ 𝜑), which is a modality sometimes called the "strong necessity" of 𝜑. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem bj-gl4

Step	Hyp	Ref	Expression
1		19.26 1878	. . . . 5 ⊢ (∀𝑥(∀𝑥𝜑 ∧ 𝜑) ↔ (∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑))
2		simpr 488	. . . . . . 7 ⊢ ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑)
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑))
4	3	anc2ri 560	. . . . 5 ⊢ (𝜑 → ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → (∀𝑥𝜑 ∧ 𝜑)))
5	1, 4	syl5bi 245	. . . 4 ⊢ (𝜑 → (∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)))
6	5	alimi 1819	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)))
7	1	biimpi 219	. . 3 ⊢ (∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑))
8	6, 7	imim12i 62	. 2 ⊢ ((∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)) → ∀𝑥(∀𝑥𝜑 ∧ 𝜑)) → (∀𝑥𝜑 → (∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑)))
9		simpl 486	. 2 ⊢ ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥∀𝑥𝜑)
10	8, 9	syl6 35	1 ⊢ ((∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)) → ∀𝑥(∀𝑥𝜑 ∧ 𝜑)) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by: (None)