Theorem vk15.4jVD 45855

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 45470 is vk15.4jVD 45855 without virtual deductions and was automatically derived from vk15.4jVD 45855. 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 45713	. . . . . 6 ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
5		vk15.4jVD.2	. . . . . . 7 ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏))
6		idn1 45516	. . . . . . . . . . . 12 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥 ¬ 𝜃   )
7		alex 1859	. . . . . . . . . . . . 13 ⊢ (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
8	7	biimpri 231	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
9	6, 8	e1a 45569	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∀𝑥𝜃   )
10		sp 2220	. . . . . . . . . . 11 ⊢ (∀𝑥𝜃 → 𝜃)
11	9, 10	e1a 45569	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝜃   )
12		idn2 45555	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜏 ∧ ¬ 𝜑)   )
13		simpl 488	. . . . . . . . . . 11 ⊢ ((𝜏 ∧ ¬ 𝜑) → 𝜏)
14	12, 13	e2 45573	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝜏   )
15		pm3.2 475	. . . . . . . . . 10 ⊢ (𝜃 → (𝜏 → (𝜃 ∧ 𝜏)))
16	11, 14, 15	e12 45665	. . . . . . . . 9 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜃 ∧ 𝜏)   )
17		19.8a 2218	. . . . . . . . 9 ⊢ ((𝜃 ∧ 𝜏) → ∃𝑥(𝜃 ∧ 𝜏))
18	16, 17	e2 45573	. . . . . . . 8 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥(𝜃 ∧ 𝜏)   )
19		notnot 143	. . . . . . . 8 ⊢ (∃𝑥(𝜃 ∧ 𝜏) → ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏))
20	18, 19	e2 45573	. . . . . . 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 45639	. . . . . 6 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ∀𝑥𝜒   )
23		hbe1 2180	. . . . . . 7 ⊢ (∃𝑥 ¬ 𝜃 → ∀𝑥∃𝑥 ¬ 𝜃)
24	23	hbn 2329	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
25		hba1 2327	. . . . . . 7 ⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒)
26	25	hbn 2329	. . . . . 6 ⊢ (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
27	4, 22, 24, 26	exinst01 45567	. . . . 5 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜒   )
28		exnal 1860	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
29	28	biimpri 231	. . . . 5 ⊢ (¬ ∀𝑥𝜒 → ∃𝑥 ¬ 𝜒)
30	27, 29	e1a 45569	. . . 4 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜒   )
31		idn2 45555	. . . . . 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 45713	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
36	12, 35	e2 45573	. . . . . . . . . . . 12 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ 𝜑   )
37		19.8a 2218	. . . . . . . . . . . 12 ⊢ (¬ 𝜑 → ∃𝑥 ¬ 𝜑)
38	36, 37	e2 45573	. . . . . . . . . . 11 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥 ¬ 𝜑   )
39		hbe1 2180	. . . . . . . . . . 11 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
40	4, 38, 24, 39	exinst01 45567	. . . . . . . . . 10 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜑   )
41		notnot 143	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
42	40, 41	e1a 45569	. . . . . . . . 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 45633	. . . . . . . 8 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)   )
45		exanali 1892	. . . . . . . . 9 ⊢ (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓 → 𝜒))
46	45	con5i 45465	. . . . . . . 8 ⊢ (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓 → 𝜒))
47	44, 46	e1a 45569	. . . . . . 7 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∀𝑥(𝜓 → 𝜒)   )
48		sp 2220	. . . . . . 7 ⊢ (∀𝑥(𝜓 → 𝜒) → (𝜓 → 𝜒))
49	47, 48	e1a 45569	. . . . . 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 45671	. . . . 5 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜓   )
53		19.8a 2218	. . . . 5 ⊢ (¬ 𝜓 → ∃𝑥 ¬ 𝜓)
54	52, 53	e2 45573	. . . 4 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶   ∃𝑥 ¬ 𝜓   )
55		hbe1 2180	. . . 4 ⊢ (∃𝑥 ¬ 𝜓 → ∀𝑥∃𝑥 ¬ 𝜓)
56	30, 54, 24, 55	exinst11 45568	. . 3 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶   ∃𝑥 ¬ 𝜓   )
57		exnal 1860	. . . 4 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
58	57	biimpi 219	. . 3 ⊢ (∃𝑥 ¬ 𝜓 → ¬ ∀𝑥𝜓)
59	56, 58	e1a 45569	. 2 ⊢ (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜓   )
60	59	in1 45513	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 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-vd1 45512  df-vd2 45520

This theorem is used by: (None)