Theorem hbexgVD 43657

Description: Virtual deduction proof of hbexg 43307. 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 43307 is hbexgVD 43657 without virtual deductions and was automatically derived from hbexgVD 43657. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

1::	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(𝜑 → ∀𝑥𝜑)   )
2:1:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 ∀𝑥(𝜑 → ∀𝑥𝜑)   )
3:2:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (𝜑 → ∀𝑥𝜑)   )
4:3:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (¬ 𝜑 → ∀𝑥¬ 𝜑)   )
5::	⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦 ∀𝑥(𝜑 → ∀𝑥𝜑))
6::	⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦 ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
7:5:	⊢ (∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
8:5,6,7:	⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦 ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
9:8,4:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 ∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑)   )
10:9:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(¬ 𝜑 → ∀𝑥¬ 𝜑)   )
11:10:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (¬ 𝜑 → ∀𝑥¬ 𝜑)   )
12:11:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   )
13:12:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀ 𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   )
14::	⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥 ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
15:13,14:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   )
16:15:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
17:16:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
18::	⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦¬ 𝜑)
19:17,18:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
20:18:	⊢ (∀𝑥∃𝑦𝜑 ↔ ∀𝑥¬ ∀𝑦¬ 𝜑)
21:19,20:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
22:8,21:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
23:14,22:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
qed:23:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )

Assertion

Ref	Expression
hbexgVD	⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Proof of Theorem hbexgVD

Step	Hyp	Ref	Expression
1		hba1 2289	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
2		hba1 2289	. . . . 5 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
3		alcom 2156	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
4	3	albii 1821	. . . . 5 ⊢ (∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
5	2, 3, 4	3imtr4i 291	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
6		idn1 43325	. . . . . . . . . . . . . . . . 17 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   )
7		ax-11 2154	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
8	6, 7	e1a 43378	. . . . . . . . . . . . . . . 16 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑)   )
9		sp 2176	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
10	8, 9	e1a 43378	. . . . . . . . . . . . . . 15 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(𝜑 → ∀𝑥𝜑)   )
11		hbntal 43304	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
12	10, 11	e1a 43378	. . . . . . . . . . . . . 14 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
13	5, 12	gen11nv 43368	. . . . . . . . . . . . 13 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
14		ax-11 2154	. . . . . . . . . . . . 13 ⊢ (∀𝑦∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
15	13, 14	e1a 43378	. . . . . . . . . . . 12 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
16		sp 2176	. . . . . . . . . . . 12 ⊢ (∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
17	15, 16	e1a 43378	. . . . . . . . . . 11 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
18		hbalg 43306	. . . . . . . . . . 11 ⊢ (∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
19	17, 18	e1a 43378	. . . . . . . . . 10 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
20		sp 2176	. . . . . . . . . 10 ⊢ (∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
21	19, 20	e1a 43378	. . . . . . . . 9 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
22	1, 21	gen11nv 43368	. . . . . . . 8 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
23		hbntal 43304	. . . . . . . 8 ⊢ (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑) → ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
24	22, 23	e1a 43378	. . . . . . 7 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
25		sp 2176	. . . . . . 7 ⊢ (∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
26	24, 25	e1a 43378	. . . . . 6 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
27		df-ex 1782	. . . . . 6 ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
28		imbi1 347	. . . . . . 7 ⊢ ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) ↔ (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
29	28	biimprcd 249	. . . . . 6 ⊢ ((¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
30	26, 27, 29	e10 43445	. . . . 5 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
31	27	albii 1821	. . . . 5 ⊢ (∀𝑥∃𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)
32		imbi2 348	. . . . . 6 ⊢ ((∀𝑥∃𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) ↔ (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
33	32	biimprcd 249	. . . . 5 ⊢ ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∀𝑥∃𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)))
34	30, 31, 33	e10 43445	. . . 4 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
35	5, 34	gen11nv 43368	. . 3 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
36	1, 35	gen11nv 43368	. 2 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
37	36	in1 43322	1 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1539  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1782  df-nf 1786  df-vd1 43321

This theorem is referenced by: (None)