Theorem vk15.4jVD 45266

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 44881 is vk15.4jVD 45266 without virtual deductions and was automatically derived from vk15.4jVD 45266. 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 1861	. . . . . . . 8 ⊢ (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏 → 𝜑))
3	2	biimpri 228	. . . . . . 7 ⊢ (¬ ∀𝑥(𝜏 → 𝜑) → ∃𝑥(𝜏 ∧ ¬ 𝜑))
4	1, 3	e0a 45124	. . . . . 6 ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
5		vk15.4jVD.2	. . . . . . 7 ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏))
6		idn1 44927	. . . . . . . . . . . 12 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥 ¬ 𝜃   )
7		alex 1828	. . . . . . . . . . . . 13 ⊢ (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
8	7	biimpri 228	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
9	6, 8	e1a 44980	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∀𝑥𝜃   )
10		sp 2191	. . . . . . . . . . 11 ⊢ (∀𝑥𝜃 → 𝜃)
11	9, 10	e1a 44980	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝜃   )
12		idn2 44966	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜏 ∧ ¬ 𝜑)   )
13		simpl 482	. . . . . . . . . . 11 ⊢ ((𝜏 ∧ ¬ 𝜑) → 𝜏)
14	12, 13	e2 44984	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝜏   )
15		pm3.2 469	. . . . . . . . . 10 ⊢ (𝜃 → (𝜏 → (𝜃 ∧ 𝜏)))
16	11, 14, 15	e12 45076	. . . . . . . . 9 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜃 ∧ 𝜏)   )
17		19.8a 2189	. . . . . . . . 9 ⊢ ((𝜃 ∧ 𝜏) → ∃𝑥(𝜃 ∧ 𝜏))
18	16, 17	e2 44984	. . . . . . . 8 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥(𝜃 ∧ 𝜏)   )
19		notnot 142	. . . . . . . 8 ⊢ (∃𝑥(𝜃 ∧ 𝜏) → ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏))
20	18, 19	e2 44984	. . . . . . 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 45050	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ∀𝑥𝜒   )
23		hbe1 2149	. . . . . . 7 ⊢ (∃𝑥 ¬ 𝜃 → ∀𝑥∃𝑥 ¬ 𝜃)
24	23	hbn 2302	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
25		hba1 2300	. . . . . . 7 ⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒)
26	25	hbn 2302	. . . . . 6 ⊢ (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
27	4, 22, 24, 26	exinst01 44978	. . . . 5 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜒   )
28		exnal 1829	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
29	28	biimpri 228	. . . . 5 ⊢ (¬ ∀𝑥𝜒 → ∃𝑥 ¬ 𝜒)
30	27, 29	e1a 44980	. . . 4 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜒   )
31		idn2 44966	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜒   )
32		vk15.4jVD.1	. . . . . . . . . 10 ⊢ ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
33		pm3.13 997	. . . . . . . . . 10 ⊢ (¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
34	32, 33	e0a 45124	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
36	12, 35	e2 44984	. . . . . . . . . . . 12 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ 𝜑   )
37		19.8a 2189	. . . . . . . . . . . 12 ⊢ (¬ 𝜑 → ∃𝑥 ¬ 𝜑)
38	36, 37	e2 44984	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥 ¬ 𝜑   )
39		hbe1 2149	. . . . . . . . . . 11 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
40	4, 38, 24, 39	exinst01 44978	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜑   )
41		notnot 142	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
42	40, 41	e1a 44980	. . . . . . . . 9 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ¬ ∃𝑥 ¬ 𝜑   )
43		pm2.53 852	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ¬ ∃𝑥 ¬ 𝜑 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
44	34, 42, 43	e01 45044	. . . . . . . 8 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)   )
45		exanali 1861	. . . . . . . . 9 ⊢ (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓 → 𝜒))
46	45	con5i 44876	. . . . . . . 8 ⊢ (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓 → 𝜒))
47	44, 46	e1a 44980	. . . . . . 7 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∀𝑥(𝜓 → 𝜒)   )
48		sp 2191	. . . . . . 7 ⊢ (∀𝑥(𝜓 → 𝜒) → (𝜓 → 𝜒))
49	47, 48	e1a 44980	. . . . . 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 45082	. . . . 5 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜓   )
53		19.8a 2189	. . . . 5 ⊢ (¬ 𝜓 → ∃𝑥 ¬ 𝜓)
54	52, 53	e2 44984	. . . 4 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶   ∃𝑥 ¬ 𝜓   )
55		hbe1 2149	. . . 4 ⊢ (∃𝑥 ¬ 𝜓 → ∀𝑥∃𝑥 ¬ 𝜓)
56	30, 54, 24, 55	exinst11 44979	. . 3 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜓   )
57		exnal 1829	. . . 4 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
58	57	biimpi 216	. . 3 ⊢ (∃𝑥 ¬ 𝜓 → ¬ ∀𝑥𝜓)
59	56, 58	e1a 44980	. 2 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜓   )
60	59	in1 44924	1 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1540  ∃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 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-vd1 44923  df-vd2 44931

This theorem is referenced by: (None)