Theorem vk15.4jVD 45744

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 45359 is vk15.4jVD 45744 without virtual deductions and was automatically derived from vk15.4jVD 45744. 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 1892	. . . . . . . 8 ⊢ (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏 → 𝜑))
3	2	biimpri 231	. . . . . . 7 ⊢ (¬ ∀𝑥(𝜏 → 𝜑) → ∃𝑥(𝜏 ∧ ¬ 𝜑))
4	1, 3	e0a 45602	. . . . . 6 ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
5		vk15.4jVD.2	. . . . . . 7 ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏))
6		idn1 45405	. . . . . . . . . . . 12 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥 ¬ 𝜃   )
7		alex 1859	. . . . . . . . . . . . 13 ⊢ (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
8	7	biimpri 231	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
9	6, 8	e1a 45458	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∀𝑥𝜃   )
10		sp 2221	. . . . . . . . . . 11 ⊢ (∀𝑥𝜃 → 𝜃)
11	9, 10	e1a 45458	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝜃   )
12		idn2 45444	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜏 ∧ ¬ 𝜑)   )
13		simpl 488	. . . . . . . . . . 11 ⊢ ((𝜏 ∧ ¬ 𝜑) → 𝜏)
14	12, 13	e2 45462	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝜏   )
15		pm3.2 475	. . . . . . . . . 10 ⊢ (𝜃 → (𝜏 → (𝜃 ∧ 𝜏)))
16	11, 14, 15	e12 45554	. . . . . . . . 9 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜃 ∧ 𝜏)   )
17		19.8a 2219	. . . . . . . . 9 ⊢ ((𝜃 ∧ 𝜏) → ∃𝑥(𝜃 ∧ 𝜏))
18	16, 17	e2 45462	. . . . . . . 8 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥(𝜃 ∧ 𝜏)   )
19		notnot 143	. . . . . . . 8 ⊢ (∃𝑥(𝜃 ∧ 𝜏) → ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏))
20	18, 19	e2 45462	. . . . . . 7 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏)   )
21		con3 154	. . . . . . 7 ⊢ ((∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏)) → (¬ ¬ ∃𝑥(𝜃 ∧ 𝜏) → ¬ ∀𝑥𝜒))
22	5, 20, 21	e02 45528	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ∀𝑥𝜒   )
23		hbe1 2180	. . . . . . 7 ⊢ (∃𝑥 ¬ 𝜃 → ∀𝑥∃𝑥 ¬ 𝜃)
24	23	hbn 2330	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
25		hba1 2328	. . . . . . 7 ⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒)
26	25	hbn 2330	. . . . . 6 ⊢ (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
27	4, 22, 24, 26	exinst01 45456	. . . . 5 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜒   )
28		exnal 1860	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
29	28	biimpri 231	. . . . 5 ⊢ (¬ ∀𝑥𝜒 → ∃𝑥 ¬ 𝜒)
30	27, 29	e1a 45458	. . . 4 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜒   )
31		idn2 45444	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜒   )
32		vk15.4jVD.1	. . . . . . . . . 10 ⊢ ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
33		pm3.13 1010	. . . . . . . . . 10 ⊢ (¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
34	32, 33	e0a 45602	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
36	12, 35	e2 45462	. . . . . . . . . . . 12 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ 𝜑   )
37		19.8a 2219	. . . . . . . . . . . 12 ⊢ (¬ 𝜑 → ∃𝑥 ¬ 𝜑)
38	36, 37	e2 45462	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥 ¬ 𝜑   )
39		hbe1 2180	. . . . . . . . . . 11 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
40	4, 38, 24, 39	exinst01 45456	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜑   )
41		notnot 143	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
42	40, 41	e1a 45458	. . . . . . . . 9 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ¬ ∃𝑥 ¬ 𝜑   )
43		pm2.53 865	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ¬ ∃𝑥 ¬ 𝜑 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
44	34, 42, 43	e01 45522	. . . . . . . 8 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)   )
45		exanali 1892	. . . . . . . . 9 ⊢ (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓 → 𝜒))
46	45	con5i 45354	. . . . . . . 8 ⊢ (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓 → 𝜒))
47	44, 46	e1a 45458	. . . . . . 7 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∀𝑥(𝜓 → 𝜒)   )
48		sp 2221	. . . . . . 7 ⊢ (∀𝑥(𝜓 → 𝜒) → (𝜓 → 𝜒))
49	47, 48	e1a 45458	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   (𝜓 → 𝜒)   )
50		con3 154	. . . . . . 7 ⊢ ((𝜓 → 𝜒) → (¬ 𝜒 → ¬ 𝜓))
51	50	com12 33	. . . . . 6 ⊢ (¬ 𝜒 → ((𝜓 → 𝜒) → ¬ 𝜓))
52	31, 49, 51	e21 45560	. . . . 5 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜓   )
53		19.8a 2219	. . . . 5 ⊢ (¬ 𝜓 → ∃𝑥 ¬ 𝜓)
54	52, 53	e2 45462	. . . 4 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶   ∃𝑥 ¬ 𝜓   )
55		hbe1 2180	. . . 4 ⊢ (∃𝑥 ¬ 𝜓 → ∀𝑥∃𝑥 ¬ 𝜓)
56	30, 54, 24, 55	exinst11 45457	. . 3 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜓   )
57		exnal 1860	. . . 4 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
58	57	biimpi 219	. . 3 ⊢ (∃𝑥 ¬ 𝜓 → ¬ ∀𝑥𝜓)
59	56, 58	e1a 45458	. 2 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜓   )
60	59	in1 45402	1 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-vd1 45401  df-vd2 45409

This theorem is used by: (None)