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Theorem hbexgVD 40779
Description: Virtual deduction proof of hbexg 40429. 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 40429 is hbexgVD 40779 without virtual deductions and was automatically derived from hbexgVD 40779. (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 2267 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑥𝑦(𝜑 → ∀𝑥𝜑))
2 hba1 2267 . . . . 5 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
3 alcom 2128 . . . . 5 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
43albii 1801 . . . . 5 (∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
52, 3, 43imtr4i 293 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑))
6 idn1 40447 . . . . . . . . . . . . . . . . 17 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦(𝜑 → ∀𝑥𝜑)   )
7 ax-11 2126 . . . . . . . . . . . . . . . . 17 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
86, 7e1a 40500 . . . . . . . . . . . . . . . 16 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝑥(𝜑 → ∀𝑥𝜑)   )
9 sp 2146 . . . . . . . . . . . . . . . 16 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
108, 9e1a 40500 . . . . . . . . . . . . . . 15 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(𝜑 → ∀𝑥𝜑)   )
11 hbntal 40426 . . . . . . . . . . . . . . 15 (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
1210, 11e1a 40500 . . . . . . . . . . . . . 14 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝜑 → ∀𝑥 ¬ 𝜑)   )
135, 12gen11nv 40490 . . . . . . . . . . . . 13 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝑥𝜑 → ∀𝑥 ¬ 𝜑)   )
14 ax-11 2126 . . . . . . . . . . . . 13 (∀𝑦𝑥𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑))
1513, 14e1a 40500 . . . . . . . . . . . 12 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑)   )
16 sp 2146 . . . . . . . . . . . 12 (∀𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦𝜑 → ∀𝑥 ¬ 𝜑))
1715, 16e1a 40500 . . . . . . . . . . 11 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝜑 → ∀𝑥 ¬ 𝜑)   )
18 hbalg 40428 . . . . . . . . . . 11 (∀𝑦𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
1917, 18e1a 40500 . . . . . . . . . 10 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
20 sp 2146 . . . . . . . . . 10 (∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
2119, 20e1a 40500 . . . . . . . . 9 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
221, 21gen11nv 40490 . . . . . . . 8 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
23 hbntal 40426 . . . . . . . 8 (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
2422, 23e1a 40500 . . . . . . 7 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
25 sp 2146 . . . . . . 7 (∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
2624, 25e1a 40500 . . . . . 6 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
27 df-ex 1762 . . . . . 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 40567 . . . . 5 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
3127albii 1801 . . . . 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 40567 . . . 4 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
355, 34gen11nv 40490 . . 3 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
361, 35gen11nv 40490 . 2 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
3736in1 40444 1 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wal 1520  wex 1761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141
This theorem depends on definitions:  df-bi 208  df-or 843  df-ex 1762  df-nf 1766  df-vd1 40443
This theorem is referenced by: (None)
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