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Theorem hbalgVD 38163
Description: Virtual deduction proof of hbalg 37792. 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 37792 is hbalgVD 38163 without virtual deductions and was automatically derived from hbalgVD 38163. (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 2137 . . 3 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦(𝜑 → ∀𝑥𝜑))
2 idn1 37811 . . . . 5 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(𝜑 → ∀𝑥𝜑)   )
3 alim 1729 . . . . 5 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
42, 3e1a 37873 . . . 4 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦𝑥𝜑)   )
5 ax-11 2021 . . . 4 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
6 imim1 81 . . . 4 ((∀𝑦𝜑 → ∀𝑦𝑥𝜑) → ((∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑)))
74, 5, 6e10 37940 . . 3 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
81, 7gen11nv 37863 . 2 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
98in1 37808 1 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701  df-vd1 37807
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator