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Theorem hbalg 39529
Description: Closed form of hbal 2210. Derived from hbalgVD 39889. (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 1906 . . 3 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
2 ax-11 2200 . . 3 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
31, 2syl6 35 . 2 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
43axc4i 2317 1 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-or 875  df-ex 1876  df-nf 1880
This theorem is referenced by:  hbexgVD  39890
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