Theorem hbalg 45284

Description: Closed form of hbal 2202. Derived from hbalgVD 45633. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbalg	⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Proof of Theorem hbalg

Step	Hyp	Ref	Expression
1		alim 1840	. . 3 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
2		ax-11 2192	. . 3 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
3	1, 2	syl6 36	. 2 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
4	3	axc4i 2355	1 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1810  df-nf 1814

This theorem is referenced by:  hbexgVD  45634