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Theorem hbexgVD 38164
Description: Virtual deduction proof of hbexg 37793. 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 37793 is hbexgVD 38164 without virtual deductions and was automatically derived from hbexgVD 38164. (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 2137 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑥𝑦(𝜑 → ∀𝑥𝜑))
2 hba1 2137 . . . . 5 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
3 alcom 2024 . . . . 5 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
43albii 1737 . . . . 5 (∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
52, 3, 43imtr4i 280 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑))
6 idn1 37811 . . . . . . . . . . . . . . . . 17 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦(𝜑 → ∀𝑥𝜑)   )
7 ax-11 2021 . . . . . . . . . . . . . . . . 17 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
86, 7e1a 37873 . . . . . . . . . . . . . . . 16 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝑥(𝜑 → ∀𝑥𝜑)   )
9 sp 2041 . . . . . . . . . . . . . . . 16 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
108, 9e1a 37873 . . . . . . . . . . . . . . 15 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(𝜑 → ∀𝑥𝜑)   )
11 hbntal 37790 . . . . . . . . . . . . . . 15 (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
1210, 11e1a 37873 . . . . . . . . . . . . . 14 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝜑 → ∀𝑥 ¬ 𝜑)   )
135, 12gen11nv 37863 . . . . . . . . . . . . 13 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝑥𝜑 → ∀𝑥 ¬ 𝜑)   )
14 ax-11 2021 . . . . . . . . . . . . 13 (∀𝑦𝑥𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑))
1513, 14e1a 37873 . . . . . . . . . . . 12 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑)   )
16 sp 2041 . . . . . . . . . . . 12 (∀𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦𝜑 → ∀𝑥 ¬ 𝜑))
1715, 16e1a 37873 . . . . . . . . . . 11 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝜑 → ∀𝑥 ¬ 𝜑)   )
18 hbalg 37792 . . . . . . . . . . 11 (∀𝑦𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
1917, 18e1a 37873 . . . . . . . . . 10 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
20 sp 2041 . . . . . . . . . 10 (∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
2119, 20e1a 37873 . . . . . . . . 9 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
221, 21gen11nv 37863 . . . . . . . 8 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
23 hbntal 37790 . . . . . . . 8 (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
2422, 23e1a 37873 . . . . . . 7 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
25 sp 2041 . . . . . . 7 (∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
2624, 25e1a 37873 . . . . . 6 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
27 df-ex 1696 . . . . . 6 (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
28 imbi1 336 . . . . . . 7 ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) ↔ (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
2928biimprcd 239 . . . . . 6 ((¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
3026, 27, 29e10 37940 . . . . 5 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
3127albii 1737 . . . . 5 (∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)
32 imbi2 337 . . . . . 6 ((∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥𝑦𝜑) ↔ (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
3332biimprcd 239 . . . . 5 ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑)))
3430, 31, 33e10 37940 . . . 4 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
355, 34gen11nv 37863 . . 3 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
361, 35gen11nv 37863 . 2 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
3736in1 37808 1 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wal 1473  wex 1695
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)
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