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Theorem vk15.4jVD 43665
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 43279 is vk15.4jVD 43665 without virtual deductions and was automatically derived from vk15.4jVD 43665. 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(𝜃𝜏 ))
h3:: ¬ ∀𝑥(𝜏𝜑)
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)   ▶   (𝜃𝜏)   )
12:11: (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝑥(𝜃𝜏)   )
13:12: (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ ¬ ∃𝑥(𝜃𝜏)   )
14:2,13: (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ ∀𝑥𝜒   )
140:: (∃𝑥¬ 𝜃 → ∀𝑥𝑥¬ 𝜃 )
141:140: (¬ ∃𝑥¬ 𝜃 → ∀𝑥¬ ∃𝑥 ¬ 𝜃)
142:: (∀𝑥𝜒 → ∀𝑥𝑥𝜒)
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 → ∀𝑥𝑥¬ 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   ▶   (𝜓𝜒 )   )
23:21,22: (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶   ¬ 𝜓   )
24:23: (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶    𝑥¬ 𝜓   )
240:: (∃𝑥¬ 𝜓 → ∀𝑥𝑥¬ 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(𝜃𝜏))
vk15.4jVD.3 ¬ ∀𝑥(𝜏𝜑)
Assertion
Ref Expression
vk15.4jVD (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Proof of Theorem vk15.4jVD
StepHypRef Expression
1 vk15.4jVD.3 . . . . . . 7 ¬ ∀𝑥(𝜏𝜑)
2 exanali 1862 . . . . . . . 8 (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏𝜑))
32biimpri 227 . . . . . . 7 (¬ ∀𝑥(𝜏𝜑) → ∃𝑥(𝜏 ∧ ¬ 𝜑))
41, 3e0a 43523 . . . . . 6 𝑥(𝜏 ∧ ¬ 𝜑)
5 vk15.4jVD.2 . . . . . . 7 (∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏))
6 idn1 43325 . . . . . . . . . . . 12 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥 ¬ 𝜃   )
7 alex 1828 . . . . . . . . . . . . 13 (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
87biimpri 227 . . . . . . . . . . . 12 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
96, 8e1a 43378 . . . . . . . . . . 11 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥𝜃   )
10 sp 2176 . . . . . . . . . . 11 (∀𝑥𝜃𝜃)
119, 10e1a 43378 . . . . . . . . . 10 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝜃   )
12 idn2 43364 . . . . . . . . . . 11 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜏 ∧ ¬ 𝜑)   )
13 simpl 483 . . . . . . . . . . 11 ((𝜏 ∧ ¬ 𝜑) → 𝜏)
1412, 13e2 43382 . . . . . . . . . 10 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝜏   )
15 pm3.2 470 . . . . . . . . . 10 (𝜃 → (𝜏 → (𝜃𝜏)))
1611, 14, 15e12 43475 . . . . . . . . 9 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜃𝜏)   )
17 19.8a 2174 . . . . . . . . 9 ((𝜃𝜏) → ∃𝑥(𝜃𝜏))
1816, 17e2 43382 . . . . . . . 8 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝑥(𝜃𝜏)   )
19 notnot 142 . . . . . . . 8 (∃𝑥(𝜃𝜏) → ¬ ¬ ∃𝑥(𝜃𝜏))
2018, 19e2 43382 . . . . . . 7 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ¬ ∃𝑥(𝜃𝜏)   )
21 con3 153 . . . . . . 7 ((∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏)) → (¬ ¬ ∃𝑥(𝜃𝜏) → ¬ ∀𝑥𝜒))
225, 20, 21e02 43448 . . . . . 6 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ∀𝑥𝜒   )
23 hbe1 2139 . . . . . . 7 (∃𝑥 ¬ 𝜃 → ∀𝑥𝑥 ¬ 𝜃)
2423hbn 2291 . . . . . 6 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
25 hba1 2289 . . . . . . 7 (∀𝑥𝜒 → ∀𝑥𝑥𝜒)
2625hbn 2291 . . . . . 6 (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
274, 22, 24, 26exinst01 43376 . . . . 5 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜒   )
28 exnal 1829 . . . . . 6 (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
2928biimpri 227 . . . . 5 (¬ ∀𝑥𝜒 → ∃𝑥 ¬ 𝜒)
3027, 29e1a 43378 . . . 4 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥 ¬ 𝜒   )
31 idn2 43364 . . . . . 6 (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜒   )
32 vk15.4jVD.1 . . . . . . . . . 10 ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
33 pm3.13 993 . . . . . . . . . 10 (¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
3432, 33e0a 43523 . . . . . . . . 9 (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35 simpr 485 . . . . . . . . . . . . 13 ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
3612, 35e2 43382 . . . . . . . . . . . 12 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ 𝜑   )
37 19.8a 2174 . . . . . . . . . . . 12 𝜑 → ∃𝑥 ¬ 𝜑)
3836, 37e2 43382 . . . . . . . . . . 11 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝑥 ¬ 𝜑   )
39 hbe1 2139 . . . . . . . . . . 11 (∃𝑥 ¬ 𝜑 → ∀𝑥𝑥 ¬ 𝜑)
404, 38, 24, 39exinst01 43376 . . . . . . . . . 10 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥 ¬ 𝜑   )
41 notnot 142 . . . . . . . . . 10 (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
4240, 41e1a 43378 . . . . . . . . 9 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ¬ ∃𝑥 ¬ 𝜑   )
43 pm2.53 849 . . . . . . . . 9 ((¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ¬ ∃𝑥 ¬ 𝜑 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
4434, 42, 43e01 43442 . . . . . . . 8 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)   )
45 exanali 1862 . . . . . . . . 9 (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓𝜒))
4645con5i 43274 . . . . . . . 8 (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓𝜒))
4744, 46e1a 43378 . . . . . . 7 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥(𝜓𝜒)   )
48 sp 2176 . . . . . . 7 (∀𝑥(𝜓𝜒) → (𝜓𝜒))
4947, 48e1a 43378 . . . . . 6 (    ¬ ∃𝑥 ¬ 𝜃   ▶   (𝜓𝜒)   )
50 con3 153 . . . . . . 7 ((𝜓𝜒) → (¬ 𝜒 → ¬ 𝜓))
5150com12 32 . . . . . 6 𝜒 → ((𝜓𝜒) → ¬ 𝜓))
5231, 49, 51e21 43481 . . . . 5 (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜓   )
53 19.8a 2174 . . . . 5 𝜓 → ∃𝑥 ¬ 𝜓)
5452, 53e2 43382 . . . 4 (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶   𝑥 ¬ 𝜓   )
55 hbe1 2139 . . . 4 (∃𝑥 ¬ 𝜓 → ∀𝑥𝑥 ¬ 𝜓)
5630, 54, 24, 55exinst11 43377 . . 3 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥 ¬ 𝜓   )
57 exnal 1829 . . . 4 (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
5857biimpi 215 . . 3 (∃𝑥 ¬ 𝜓 → ¬ ∀𝑥𝜓)
5956, 58e1a 43378 . 2 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜓   )
6059in1 43322 1 (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845  wal 1539  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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-vd1 43321  df-vd2 43329
This theorem is referenced by: (None)
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