Theorem hbexgVD 42140

Description: Virtual deduction proof of hbexg 41790. 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 41790 is hbexgVD 42140 without virtual deductions and was automatically derived from hbexgVD 42140. (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 2296	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
2		hba1 2296	. . . . 5 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
3		alcom 2162	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
4	3	albii 1827	. . . . 5 ⊢ (∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
5	2, 3, 4	3imtr4i 295	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
6		idn1 41808	. . . . . . . . . . . . . . . . 17 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   )
7		ax-11 2160	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
8	6, 7	e1a 41861	. . . . . . . . . . . . . . . 16 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑)   )
9		sp 2182	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
10	8, 9	e1a 41861	. . . . . . . . . . . . . . 15 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(𝜑 → ∀𝑥𝜑)   )
11		hbntal 41787	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
12	10, 11	e1a 41861	. . . . . . . . . . . . . 14 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
13	5, 12	gen11nv 41851	. . . . . . . . . . . . 13 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
14		ax-11 2160	. . . . . . . . . . . . 13 ⊢ (∀𝑦∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
15	13, 14	e1a 41861	. . . . . . . . . . . 12 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
16		sp 2182	. . . . . . . . . . . 12 ⊢ (∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
17	15, 16	e1a 41861	. . . . . . . . . . 11 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
18		hbalg 41789	. . . . . . . . . . 11 ⊢ (∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
19	17, 18	e1a 41861	. . . . . . . . . 10 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
20		sp 2182	. . . . . . . . . 10 ⊢ (∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
21	19, 20	e1a 41861	. . . . . . . . 9 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
22	1, 21	gen11nv 41851	. . . . . . . 8 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
23		hbntal 41787	. . . . . . . 8 ⊢ (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑) → ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
24	22, 23	e1a 41861	. . . . . . 7 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
25		sp 2182	. . . . . . 7 ⊢ (∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
26	24, 25	e1a 41861	. . . . . 6 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
27		df-ex 1788	. . . . . 6 ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
28		imbi1 351	. . . . . . 7 ⊢ ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) ↔ (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
29	28	biimprcd 253	. . . . . 6 ⊢ ((¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
30	26, 27, 29	e10 41928	. . . . 5 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
31	27	albii 1827	. . . . 5 ⊢ (∀𝑥∃𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)
32		imbi2 352	. . . . . 6 ⊢ ((∀𝑥∃𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) ↔ (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
33	32	biimprcd 253	. . . . 5 ⊢ ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∀𝑥∃𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)))
34	30, 31, 33	e10 41928	. . . 4 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
35	5, 34	gen11nv 41851	. . 3 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
36	1, 35	gen11nv 41851	. 2 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
37	36	in1 41805	1 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-11 2160  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-or 848  df-ex 1788  df-nf 1792  df-vd1 41804

This theorem is referenced by: (None)