Theorem bj-hbalt 35000

Description: Closed form of hbal 2167. When in main part, prove hbal 2167 and hbald 2168 from it. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-hbalt	⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Proof of Theorem bj-hbalt

Step	Hyp	Ref	Expression
1		alim 1812	. 2 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
2		ax-11 2154	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
3	1, 2	syl6 35	1 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-4 1811  ax-11 2154

This theorem is referenced by:  bj-hbext  35029  bj-nfalt  35030  bj-cbv3ta  35105