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Theorem hbalgVD 37962
Description: Virtual deduction proof of hbalg 37591. 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 37591 is hbalgVD 37962 without virtual deductions and was automatically derived from hbalgVD 37962. (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 2134 . . 3 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦(𝜑 → ∀𝑥𝜑))
2 idn1 37610 . . . . 5 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(𝜑 → ∀𝑥𝜑)   )
3 alim 1727 . . . . 5 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
42, 3e1a 37672 . . . 4 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦𝑥𝜑)   )
5 ax-11 2019 . . . 4 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
6 imim1 80 . . . 4 ((∀𝑦𝜑 → ∀𝑦𝑥𝜑) → ((∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑)))
74, 5, 6e10 37739 . . 3 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
81, 7gen11nv 37662 . 2 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
98in1 37607 1 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695  df-nf 1700  df-vd1 37606
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator