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Theorem hbalgVD 43437
Description: Virtual deduction proof of hbalg 43087. 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 43087 is hbalgVD 43437 without virtual deductions and was automatically derived from hbalgVD 43437. (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 → ∀𝑦𝑥𝜑)   )
3:: (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
4:2,3: (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
5:: (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦( 𝜑 → ∀𝑥𝜑))
6:5,4: (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀ 𝑦𝜑 → ∀𝑥𝑦𝜑)   )
qed:6: (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦 𝜑 → ∀𝑥𝑦𝜑))
Assertion
Ref Expression
hbalgVD (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Proof of Theorem hbalgVD
StepHypRef Expression
1 hba1 2289 . . 3 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦(𝜑 → ∀𝑥𝜑))
2 idn1 43106 . . . . 5 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(𝜑 → ∀𝑥𝜑)   )
3 alim 1812 . . . . 5 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
42, 3e1a 43159 . . . 4 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦𝑥𝜑)   )
5 ax-11 2154 . . . 4 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
6 imim1 83 . . . 4 ((∀𝑦𝜑 → ∀𝑦𝑥𝜑) → ((∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑)))
74, 5, 6e10 43226 . . 3 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
81, 7gen11nv 43149 . 2 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
98in1 43103 1 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539
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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1782  df-nf 1786  df-vd1 43102
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator