Theorem bj-hbalt 15699

Description: Closed form of hbal 1500 (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 1480	. 2 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
2		ax-7 1471	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
3	1, 2	syl6 33	1 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1470  ax-7 1471

This theorem is referenced by:  bj-nfalt  15700