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Theorem hbexgVD 41117
Description: Virtual deduction proof of hbexg 40767. 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. hbexg 40767 is hbexgVD 41117 without virtual deductions and was automatically derived from hbexgVD 41117. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(𝜑 → ∀𝑥𝜑)   )
2:1: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝑥(𝜑 → ∀𝑥𝜑)   )
3:2: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (𝜑 → ∀𝑥𝜑)   )
4:3: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝜑 → ∀𝑥¬ 𝜑)   )
5:: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦 𝑥(𝜑 → ∀𝑥𝜑))
6:: (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦 𝑦𝑥(𝜑 → ∀𝑥𝜑))
7:5: (∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ 𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
8:5,6,7: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦 𝑥𝑦(𝜑 → ∀𝑥𝜑))
9:8,4: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝑥𝜑 → ∀𝑥¬ 𝜑)   )
10:9: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦𝜑 → ∀𝑥¬ 𝜑)   )
11:10: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝜑 → ∀𝑥¬ 𝜑)   )
12:11: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 (∀𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
13:12: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀ 𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
14:: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥 𝑥𝑦(𝜑 → ∀𝑥𝜑))
15:13,14: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (∀𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
16:15: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
17:16: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶    𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
18:: (∃𝑦𝜑 ↔ ¬ ∀𝑦¬ 𝜑)
19:17,18: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
20:18: (∀𝑥𝑦𝜑 ↔ ∀𝑥¬ ∀𝑦¬ 𝜑)
21:19,20: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥𝑦𝜑)   )
22:8,21: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
23:14,22: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
qed:23: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
Assertion
Ref Expression
hbexgVD (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Proof of Theorem hbexgVD
StepHypRef Expression
1 hba1 2292 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑥𝑦(𝜑 → ∀𝑥𝜑))
2 hba1 2292 . . . . 5 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
3 alcom 2153 . . . . 5 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
43albii 1811 . . . . 5 (∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
52, 3, 43imtr4i 293 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑))
6 idn1 40785 . . . . . . . . . . . . . . . . 17 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦(𝜑 → ∀𝑥𝜑)   )
7 ax-11 2151 . . . . . . . . . . . . . . . . 17 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
86, 7e1a 40838 . . . . . . . . . . . . . . . 16 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝑥(𝜑 → ∀𝑥𝜑)   )
9 sp 2172 . . . . . . . . . . . . . . . 16 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
108, 9e1a 40838 . . . . . . . . . . . . . . 15 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(𝜑 → ∀𝑥𝜑)   )
11 hbntal 40764 . . . . . . . . . . . . . . 15 (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
1210, 11e1a 40838 . . . . . . . . . . . . . 14 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝜑 → ∀𝑥 ¬ 𝜑)   )
135, 12gen11nv 40828 . . . . . . . . . . . . 13 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝑥𝜑 → ∀𝑥 ¬ 𝜑)   )
14 ax-11 2151 . . . . . . . . . . . . 13 (∀𝑦𝑥𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑))
1513, 14e1a 40838 . . . . . . . . . . . 12 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑)   )
16 sp 2172 . . . . . . . . . . . 12 (∀𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦𝜑 → ∀𝑥 ¬ 𝜑))
1715, 16e1a 40838 . . . . . . . . . . 11 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝜑 → ∀𝑥 ¬ 𝜑)   )
18 hbalg 40766 . . . . . . . . . . 11 (∀𝑦𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
1917, 18e1a 40838 . . . . . . . . . 10 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
20 sp 2172 . . . . . . . . . 10 (∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
2119, 20e1a 40838 . . . . . . . . 9 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
221, 21gen11nv 40828 . . . . . . . 8 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
23 hbntal 40764 . . . . . . . 8 (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
2422, 23e1a 40838 . . . . . . 7 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
25 sp 2172 . . . . . . 7 (∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
2624, 25e1a 40838 . . . . . 6 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
27 df-ex 1772 . . . . . 6 (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
28 imbi1 349 . . . . . . 7 ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) ↔ (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
2928biimprcd 251 . . . . . 6 ((¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
3026, 27, 29e10 40905 . . . . 5 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
3127albii 1811 . . . . 5 (∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)
32 imbi2 350 . . . . . 6 ((∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥𝑦𝜑) ↔ (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
3332biimprcd 251 . . . . 5 ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑)))
3430, 31, 33e10 40905 . . . 4 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
355, 34gen11nv 40828 . . 3 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
361, 35gen11nv 40828 . 2 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
3736in1 40782 1 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-or 842  df-ex 1772  df-nf 1776  df-vd1 40781
This theorem is referenced by: (None)
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