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Theorem hbalgVD 42854
Description: Virtual deduction proof of hbalg 42504. 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 42504 is hbalgVD 42854 without virtual deductions and was automatically derived from hbalgVD 42854. (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 42523 . . . . 5 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(𝜑 → ∀𝑥𝜑)   )
3 alim 1811 . . . . 5 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
42, 3e1a 42576 . . . 4 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦𝑥𝜑)   )
5 ax-11 2153 . . . 4 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
6 imim1 83 . . . 4 ((∀𝑦𝜑 → ∀𝑦𝑥𝜑) → ((∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑)))
74, 5, 6e10 42643 . . 3 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
81, 7gen11nv 42566 . 2 (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
98in1 42520 1 (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-11 2153  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1781  df-nf 1785  df-vd1 42519
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator