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Theorem hbalg 37591
Description: Closed form of hbal 2021. Derived from hbalgVD 37962. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbalg (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Proof of Theorem hbalg
StepHypRef Expression
1 alim 1727 . . 3 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
2 ax-11 2019 . . 3 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
31, 2syl6 34 . 2 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
43axc4i 2113 1 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀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-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695  df-nf 1700
This theorem is referenced by:  hbexgVD  37963
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