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Theorem bj-hbalt 31651
Description: Closed form of hbal 2022. When in main part, prove hbal 2022 and hbald 2027 from it. (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-hbalt (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Proof of Theorem bj-hbalt
StepHypRef Expression
1 alim 1728 . 2 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
2 ax-11 2020 . 2 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
31, 2syl6 34 1 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-4 1727  ax-11 2020
This theorem is referenced by:  bj-hbext  31681  bj-nfalt  31682  bj-cbv3ta  31690
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