Theorem bj-hbalt 31858

Description: Closed form of hbal 2023. When in main part, prove hbal 2023 and hbald 2028 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 1729	. 2 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
2		ax-11 2021	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
3	1, 2	syl6 34	1 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-4 1728  ax-11 2021

This theorem is referenced by:  bj-hbext  31888  bj-nfalt  31889  bj-cbv3ta  31897