Theorem hbalgVD 46026

Description: Virtual deduction proof of hbalg 45677. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbalg 45677 is hbalgVD 46026 without virtual deductions and was automatically derived from hbalgVD 46026. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

1::	⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(𝜑 → ∀𝑥𝜑)   )
2:1:	⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)   )
3::	⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
4:2,3:	⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)   )
5::	⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦( 𝜑 → ∀𝑥𝜑))
6:5,4:	⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∀ 𝑦𝜑 → ∀𝑥∀𝑦𝜑)   )
qed:6:	⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦 𝜑 → ∀𝑥∀𝑦𝜑))

Assertion

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

Proof of Theorem hbalgVD

Step	Hyp	Ref	Expression
1		hba1 2329	. . 3 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦(𝜑 → ∀𝑥𝜑))
2		idn1 45696	. . . . 5 ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(𝜑 → ∀𝑥𝜑)   )
3		alim 1842	. . . . 5 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
4	2, 3	e1a 45749	. . . 4 ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)   )
5		ax-11 2196	. . . 4 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
6		imim1 84	. . . 4 ⊢ ((∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) → ((∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)))
7	4, 5, 6	e10 45816	. . 3 ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)   )
8	1, 7	gen11nv 45739	. 2 ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)   )
9	8	in1 45693	1 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-10 2179  ax-11 2196  ax-12 2215

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816  df-vd1 45692

This theorem is used by: (None)