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Theorem hbalgVD 38663
Description: Virtual deduction proof of hbalg 38292. 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 38292 is hbalgVD 38663 without virtual deductions and was automatically derived from hbalgVD 38663. (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 2148 . . 3 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦(𝜑 → ∀𝑥𝜑))
2 idn1 38311 . . . . 5 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(𝜑 → ∀𝑥𝜑)   )
3 alim 1735 . . . . 5 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
42, 3e1a 38373 . . . 4 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦𝑥𝜑)   )
5 ax-11 2031 . . . 4 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
6 imim1 83 . . . 4 ((∀𝑦𝜑 → ∀𝑦𝑥𝜑) → ((∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑)))
74, 5, 6e10 38440 . . 3 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
81, 7gen11nv 38363 . 2 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
98in1 38308 1 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707  df-vd1 38307
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator