Theorem bj-gl4lem 32579

Description: Lemma for bj-gl4 32580. Note that this proof holds in the modal logic (K). (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-gl4lem	⊢ (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)))

Proof of Theorem bj-gl4lem

Step	Hyp	Ref	Expression
1		19.26 1798	. . 3 ⊢ (∀𝑥(∀𝑥𝜑 ∧ 𝜑) ↔ (∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑))
2		simpr 477	. . . . 5 ⊢ ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑)
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑))
4	3	anc2ri 581	. . 3 ⊢ (𝜑 → ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → (∀𝑥𝜑 ∧ 𝜑)))
5	1, 4	syl5bi 232	. 2 ⊢ (𝜑 → (∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)))
6	5	alimi 1739	1 ⊢ (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  bj-gl4  32580