Theorem hbalgVD 45352

Description: Virtual deduction proof of hbalg 45003. 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 45003 is hbalgVD 45352 without virtual deductions and was automatically derived from hbalgVD 45352. (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 2300	. . 3 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦(𝜑 → ∀𝑥𝜑))
2		idn1 45022	. . . . 5 ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(𝜑 → ∀𝑥𝜑)   )
3		alim 1812	. . . . 5 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
4	2, 3	e1a 45075	. . . 4 ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)   )
5		ax-11 2163	. . . 4 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
6		imim1 83	. . . 4 ⊢ ((∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) → ((∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)))
7	4, 5, 6	e10 45142	. . 3 ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)   )
8	1, 7	gen11nv 45065	. 2 ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)   )
9	8	in1 45019	1 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1782  df-nf 1786  df-vd1 45018

This theorem is referenced by: (None)