Theorem bj-hbalt 16791

Description: Closed form of hbal 1530 (copied from set.mm). (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 1510	. 2 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
2		ax-7 1501	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
3	1, 2	syl6 33	1 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1500  ax-7 1501

This theorem is used by:  bj-nfalt  16792