Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-gl4 Structured version   Visualization version   GIF version

Theorem bj-gl4 34058
 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 34058 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(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)) → ∀𝑥(∀𝑥𝜑𝜑)) → (∀𝑥𝜑 → ∀𝑥𝑥𝜑))

Proof of Theorem bj-gl4
StepHypRef Expression
1 19.26 1871 . . . . 5 (∀𝑥(∀𝑥𝜑𝜑) ↔ (∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑))
2 simpr 488 . . . . . . 7 ((∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑)
32a1i 11 . . . . . 6 (𝜑 → ((∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑))
43anc2ri 560 . . . . 5 (𝜑 → ((∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑) → (∀𝑥𝜑𝜑)))
51, 4syl5bi 245 . . . 4 (𝜑 → (∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))
65alimi 1813 . . 3 (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))
71biimpi 219 . . 3 (∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑))
86, 7imim12i 62 . 2 ((∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)) → ∀𝑥(∀𝑥𝜑𝜑)) → (∀𝑥𝜑 → (∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑)))
9 simpl 486 . 2 ((∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝑥𝜑)
108, 9syl6 35 1 ((∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)) → ∀𝑥(∀𝑥𝜑𝜑)) → (∀𝑥𝜑 → ∀𝑥𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator