Theorem vk15.4jVD 44905

Description: The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Unit 15 Excercise 4.f. found in the "Answers to Starred Exercises" on page 442 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. The same proof may also be interpreted to be a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. vk15.4j 44520 is vk15.4jVD 44905 without virtual deductions and was automatically derived from vk15.4jVD 44905. Step numbers greater than 25 are additional steps necessary for the sequent calculus proof not contained in the Fitch-style proof. Otherwise, step i of the User's Proof corresponds to step i of the Fitch-style proof.

h1::	⊢ ¬ (∃𝑥¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
h2::	⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏 ))
h3::	⊢ ¬ ∀𝑥(𝜏 → 𝜑)
4::	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ∃𝑥¬ 𝜃   )
5:4:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∀𝑥𝜃   )
6:3:	⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
7::	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜏 ∧ ¬ 𝜑)   )
8:7:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝜏   )
9:7:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ 𝜑   )
10:5:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   𝜃   )
11:10,8:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜃 ∧ 𝜏)   )
12:11:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥(𝜃 ∧ 𝜏)   )
13:12:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏)   )
14:2,13:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ ∀𝑥𝜒   )
140::	⊢ (∃𝑥¬ 𝜃 → ∀𝑥∃𝑥¬ 𝜃 )
141:140:	⊢ (¬ ∃𝑥¬ 𝜃 → ∀𝑥¬ ∃𝑥 ¬ 𝜃)
142::	⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒)
143:142:	⊢ (¬ ∀𝑥𝜒 → ∀𝑥¬ ∀𝑥𝜒 )
144:6,14,141,143:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ∀𝑥𝜒    )
15:1:	⊢ (¬ ∃𝑥¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
16:9:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥¬ 𝜑   )
161::	⊢ (∃𝑥¬ 𝜑 → ∀𝑥∃𝑥¬ 𝜑 )
162:6,16,141,161:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∃𝑥¬ 𝜑    )
17:162:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ¬ ∃𝑥 ¬ 𝜑   )
18:15,17:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ∃𝑥( 𝜓 ∧ ¬ 𝜒)   )
19:18:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∀𝑥(𝜓 → 𝜒)   )
20:144:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∃𝑥¬ 𝜒    )
21::	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶   ¬ 𝜒   )
22:19:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   (𝜓 → 𝜒 )   )
23:21,22:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶   ¬ 𝜓   )
24:23:	⊢ (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶   ∃ 𝑥¬ 𝜓   )
240::	⊢ (∃𝑥¬ 𝜓 → ∀𝑥∃𝑥¬ 𝜓 )
241:20,24,141,240:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∃𝑥¬ 𝜓    )
25:241:	⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ∀𝑥𝜓    )
qed:25:	⊢ (¬ ∃𝑥¬ 𝜃 → ¬ ∀𝑥𝜓)

(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vk15.4jVD.1	⊢ ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
vk15.4jVD.2	⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏))
vk15.4jVD.3	⊢ ¬ ∀𝑥(𝜏 → 𝜑)

Assertion

Ref	Expression
vk15.4jVD	⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Proof of Theorem vk15.4jVD

Step	Hyp	Ref	Expression
1		vk15.4jVD.3	. . . . . . 7 ⊢ ¬ ∀𝑥(𝜏 → 𝜑)
2		exanali 1859	. . . . . . . 8 ⊢ (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏 → 𝜑))
3	2	biimpri 228	. . . . . . 7 ⊢ (¬ ∀𝑥(𝜏 → 𝜑) → ∃𝑥(𝜏 ∧ ¬ 𝜑))
4	1, 3	e0a 44763	. . . . . 6 ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
5		vk15.4jVD.2	. . . . . . 7 ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏))
6		idn1 44566	. . . . . . . . . . . 12 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥 ¬ 𝜃   )
7		alex 1826	. . . . . . . . . . . . 13 ⊢ (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
8	7	biimpri 228	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
9	6, 8	e1a 44619	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∀𝑥𝜃   )
10		sp 2184	. . . . . . . . . . 11 ⊢ (∀𝑥𝜃 → 𝜃)
11	9, 10	e1a 44619	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝜃   )
12		idn2 44605	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜏 ∧ ¬ 𝜑)   )
13		simpl 482	. . . . . . . . . . 11 ⊢ ((𝜏 ∧ ¬ 𝜑) → 𝜏)
14	12, 13	e2 44623	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝜏   )
15		pm3.2 469	. . . . . . . . . 10 ⊢ (𝜃 → (𝜏 → (𝜃 ∧ 𝜏)))
16	11, 14, 15	e12 44715	. . . . . . . . 9 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜃 ∧ 𝜏)   )
17		19.8a 2182	. . . . . . . . 9 ⊢ ((𝜃 ∧ 𝜏) → ∃𝑥(𝜃 ∧ 𝜏))
18	16, 17	e2 44623	. . . . . . . 8 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥(𝜃 ∧ 𝜏)   )
19		notnot 142	. . . . . . . 8 ⊢ (∃𝑥(𝜃 ∧ 𝜏) → ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏))
20	18, 19	e2 44623	. . . . . . 7 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏)   )
21		con3 153	. . . . . . 7 ⊢ ((∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏)) → (¬ ¬ ∃𝑥(𝜃 ∧ 𝜏) → ¬ ∀𝑥𝜒))
22	5, 20, 21	e02 44689	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ∀𝑥𝜒   )
23		hbe1 2144	. . . . . . 7 ⊢ (∃𝑥 ¬ 𝜃 → ∀𝑥∃𝑥 ¬ 𝜃)
24	23	hbn 2296	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
25		hba1 2294	. . . . . . 7 ⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒)
26	25	hbn 2296	. . . . . 6 ⊢ (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
27	4, 22, 24, 26	exinst01 44617	. . . . 5 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜒   )
28		exnal 1827	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
29	28	biimpri 228	. . . . 5 ⊢ (¬ ∀𝑥𝜒 → ∃𝑥 ¬ 𝜒)
30	27, 29	e1a 44619	. . . 4 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜒   )
31		idn2 44605	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜒   )
32		vk15.4jVD.1	. . . . . . . . . 10 ⊢ ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
33		pm3.13 996	. . . . . . . . . 10 ⊢ (¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
34	32, 33	e0a 44763	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
36	12, 35	e2 44623	. . . . . . . . . . . 12 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ 𝜑   )
37		19.8a 2182	. . . . . . . . . . . 12 ⊢ (¬ 𝜑 → ∃𝑥 ¬ 𝜑)
38	36, 37	e2 44623	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥 ¬ 𝜑   )
39		hbe1 2144	. . . . . . . . . . 11 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
40	4, 38, 24, 39	exinst01 44617	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜑   )
41		notnot 142	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
42	40, 41	e1a 44619	. . . . . . . . 9 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ¬ ∃𝑥 ¬ 𝜑   )
43		pm2.53 851	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ¬ ∃𝑥 ¬ 𝜑 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
44	34, 42, 43	e01 44683	. . . . . . . 8 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)   )
45		exanali 1859	. . . . . . . . 9 ⊢ (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓 → 𝜒))
46	45	con5i 44515	. . . . . . . 8 ⊢ (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓 → 𝜒))
47	44, 46	e1a 44619	. . . . . . 7 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∀𝑥(𝜓 → 𝜒)   )
48		sp 2184	. . . . . . 7 ⊢ (∀𝑥(𝜓 → 𝜒) → (𝜓 → 𝜒))
49	47, 48	e1a 44619	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   (𝜓 → 𝜒)   )
50		con3 153	. . . . . . 7 ⊢ ((𝜓 → 𝜒) → (¬ 𝜒 → ¬ 𝜓))
51	50	com12 32	. . . . . 6 ⊢ (¬ 𝜒 → ((𝜓 → 𝜒) → ¬ 𝜓))
52	31, 49, 51	e21 44721	. . . . 5 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜓   )
53		19.8a 2182	. . . . 5 ⊢ (¬ 𝜓 → ∃𝑥 ¬ 𝜓)
54	52, 53	e2 44623	. . . 4 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶   ∃𝑥 ¬ 𝜓   )
55		hbe1 2144	. . . 4 ⊢ (∃𝑥 ¬ 𝜓 → ∀𝑥∃𝑥 ¬ 𝜓)
56	30, 54, 24, 55	exinst11 44618	. . 3 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜓   )
57		exnal 1827	. . . 4 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
58	57	biimpi 216	. . 3 ⊢ (∃𝑥 ¬ 𝜓 → ¬ ∀𝑥𝜓)
59	56, 58	e1a 44619	. 2 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜓   )
60	59	in1 44563	1 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847  ∀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-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-vd1 44562  df-vd2 44570

This theorem is referenced by: (None)