Theorem bj-gl4 31753

Description: In a normal modal logic, the modal axiom GL implies the modal axiom (4). Note that the antecedent of bj-gl4 31753 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑 ∧ 𝜑), 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		bj-gl4lem 31752	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)))
2		19.26 1786	. . . 4 ⊢ (∀𝑥(∀𝑥𝜑 ∧ 𝜑) ↔ (∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑))
3	2	biimpi 205	. . 3 ⊢ (∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑))
4	1, 3	imim12i 60	. 2 ⊢ ((∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)) → ∀𝑥(∀𝑥𝜑 ∧ 𝜑)) → (∀𝑥𝜑 → (∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑)))
5		simpl 472	. 2 ⊢ ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥∀𝑥𝜑)
6	4, 5	syl6 34	1 ⊢ ((∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)) → ∀𝑥(∀𝑥𝜑 ∧ 𝜑)) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385

This theorem is referenced by: (None)