Theorem hbexgVD 44905

Description: Virtual deduction proof of hbexg 44556. 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 44556 is hbexgVD 44905 without virtual deductions and was automatically derived from hbexgVD 44905. (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 2294	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
2		hba1 2294	. . . . 5 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
3		alcom 2160	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
4	3	albii 1819	. . . . 5 ⊢ (∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
5	2, 3, 4	3imtr4i 292	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
6		idn1 44574	. . . . . . . . . . . . . . . . 17 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   )
7		ax-11 2158	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
8	6, 7	e1a 44627	. . . . . . . . . . . . . . . 16 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑)   )
9		sp 2184	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
10	8, 9	e1a 44627	. . . . . . . . . . . . . . 15 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(𝜑 → ∀𝑥𝜑)   )
11		hbntal 44553	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
12	10, 11	e1a 44627	. . . . . . . . . . . . . 14 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
13	5, 12	gen11nv 44617	. . . . . . . . . . . . 13 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
14		ax-11 2158	. . . . . . . . . . . . 13 ⊢ (∀𝑦∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
15	13, 14	e1a 44627	. . . . . . . . . . . 12 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
16		sp 2184	. . . . . . . . . . . 12 ⊢ (∀𝑥∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
17	15, 16	e1a 44627	. . . . . . . . . . 11 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑)   )
18		hbalg 44555	. . . . . . . . . . 11 ⊢ (∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
19	17, 18	e1a 44627	. . . . . . . . . 10 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
20		sp 2184	. . . . . . . . . 10 ⊢ (∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
21	19, 20	e1a 44627	. . . . . . . . 9 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
22	1, 21	gen11nv 44617	. . . . . . . 8 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)   )
23		hbntal 44553	. . . . . . . 8 ⊢ (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑) → ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
24	22, 23	e1a 44627	. . . . . . 7 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
25		sp 2184	. . . . . . 7 ⊢ (∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
26	24, 25	e1a 44627	. . . . . 6 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
27		df-ex 1780	. . . . . 6 ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
28		imbi1 347	. . . . . . 7 ⊢ ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) ↔ (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
29	28	biimprcd 250	. . . . . 6 ⊢ ((¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
30	26, 27, 29	e10 44694	. . . . 5 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
31	27	albii 1819	. . . . 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 250	. . . . 5 ⊢ ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∀𝑥∃𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)))
34	30, 31, 33	e10 44694	. . . 4 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
35	5, 34	gen11nv 44617	. . 3 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
36	1, 35	gen11nv 44617	. 2 ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
37	36	in1 44571	1 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1780  df-nf 1784  df-vd1 44570

This theorem is referenced by: (None)