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Theorem vk15.4jVD 43977
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 43591 is vk15.4jVD 43977 without virtual deductions and was automatically derived from vk15.4jVD 43977. 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 1860 . . . . . . . 8 (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏𝜑))
32biimpri 227 . . . . . . 7 (¬ ∀𝑥(𝜏𝜑) → ∃𝑥(𝜏 ∧ ¬ 𝜑))
41, 3e0a 43835 . . . . . 6 𝑥(𝜏 ∧ ¬ 𝜑)
5 vk15.4jVD.2 . . . . . . 7 (∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏))
6 idn1 43637 . . . . . . . . . . . 12 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥 ¬ 𝜃   )
7 alex 1826 . . . . . . . . . . . . 13 (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
87biimpri 227 . . . . . . . . . . . 12 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
96, 8e1a 43690 . . . . . . . . . . 11 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥𝜃   )
10 sp 2174 . . . . . . . . . . 11 (∀𝑥𝜃𝜃)
119, 10e1a 43690 . . . . . . . . . 10 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝜃   )
12 idn2 43676 . . . . . . . . . . 11 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜏 ∧ ¬ 𝜑)   )
13 simpl 481 . . . . . . . . . . 11 ((𝜏 ∧ ¬ 𝜑) → 𝜏)
1412, 13e2 43694 . . . . . . . . . 10 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝜏   )
15 pm3.2 468 . . . . . . . . . 10 (𝜃 → (𝜏 → (𝜃𝜏)))
1611, 14, 15e12 43787 . . . . . . . . 9 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜃𝜏)   )
17 19.8a 2172 . . . . . . . . 9 ((𝜃𝜏) → ∃𝑥(𝜃𝜏))
1816, 17e2 43694 . . . . . . . 8 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝑥(𝜃𝜏)   )
19 notnot 142 . . . . . . . 8 (∃𝑥(𝜃𝜏) → ¬ ¬ ∃𝑥(𝜃𝜏))
2018, 19e2 43694 . . . . . . 7 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ¬ ∃𝑥(𝜃𝜏)   )
21 con3 153 . . . . . . 7 ((∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏)) → (¬ ¬ ∃𝑥(𝜃𝜏) → ¬ ∀𝑥𝜒))
225, 20, 21e02 43760 . . . . . 6 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ ∀𝑥𝜒   )
23 hbe1 2137 . . . . . . 7 (∃𝑥 ¬ 𝜃 → ∀𝑥𝑥 ¬ 𝜃)
2423hbn 2289 . . . . . 6 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
25 hba1 2287 . . . . . . 7 (∀𝑥𝜒 → ∀𝑥𝑥𝜒)
2625hbn 2289 . . . . . 6 (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
274, 22, 24, 26exinst01 43688 . . . . 5 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜒   )
28 exnal 1827 . . . . . 6 (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
2928biimpri 227 . . . . 5 (¬ ∀𝑥𝜒 → ∃𝑥 ¬ 𝜒)
3027, 29e1a 43690 . . . 4 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥 ¬ 𝜒   )
31 idn2 43676 . . . . . 6 (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜒   )
32 vk15.4jVD.1 . . . . . . . . . 10 ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
33 pm3.13 991 . . . . . . . . . 10 (¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
3432, 33e0a 43835 . . . . . . . . 9 (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35 simpr 483 . . . . . . . . . . . . 13 ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
3612, 35e2 43694 . . . . . . . . . . . 12 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶    ¬ 𝜑   )
37 19.8a 2172 . . . . . . . . . . . 12 𝜑 → ∃𝑥 ¬ 𝜑)
3836, 37e2 43694 . . . . . . . . . . 11 (    ¬ ∃𝑥 ¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝑥 ¬ 𝜑   )
39 hbe1 2137 . . . . . . . . . . 11 (∃𝑥 ¬ 𝜑 → ∀𝑥𝑥 ¬ 𝜑)
404, 38, 24, 39exinst01 43688 . . . . . . . . . 10 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥 ¬ 𝜑   )
41 notnot 142 . . . . . . . . . 10 (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
4240, 41e1a 43690 . . . . . . . . 9 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ¬ ∃𝑥 ¬ 𝜑   )
43 pm2.53 847 . . . . . . . . 9 ((¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ¬ ∃𝑥 ¬ 𝜑 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
4434, 42, 43e01 43754 . . . . . . . 8 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)   )
45 exanali 1860 . . . . . . . . 9 (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓𝜒))
4645con5i 43586 . . . . . . . 8 (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓𝜒))
4744, 46e1a 43690 . . . . . . 7 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥(𝜓𝜒)   )
48 sp 2174 . . . . . . 7 (∀𝑥(𝜓𝜒) → (𝜓𝜒))
4947, 48e1a 43690 . . . . . 6 (    ¬ ∃𝑥 ¬ 𝜃   ▶   (𝜓𝜒)   )
50 con3 153 . . . . . . 7 ((𝜓𝜒) → (¬ 𝜒 → ¬ 𝜓))
5150com12 32 . . . . . 6 𝜒 → ((𝜓𝜒) → ¬ 𝜓))
5231, 49, 51e21 43793 . . . . 5 (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶    ¬ 𝜓   )
53 19.8a 2172 . . . . 5 𝜓 → ∃𝑥 ¬ 𝜓)
5452, 53e2 43694 . . . 4 (    ¬ ∃𝑥 ¬ 𝜃   ,    ¬ 𝜒   ▶   𝑥 ¬ 𝜓   )
55 hbe1 2137 . . . 4 (∃𝑥 ¬ 𝜓 → ∀𝑥𝑥 ¬ 𝜓)
5630, 54, 24, 55exinst11 43689 . . 3 (    ¬ ∃𝑥 ¬ 𝜃   ▶   𝑥 ¬ 𝜓   )
57 exnal 1827 . . . 4 (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
5857biimpi 215 . . 3 (∃𝑥 ¬ 𝜓 → ¬ ∀𝑥𝜓)
5956, 58e1a 43690 . 2 (    ¬ ∃𝑥 ¬ 𝜃   ▶    ¬ ∀𝑥𝜓   )
6059in1 43634 1 (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 843  wal 1537  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 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ex 1780  df-nf 1784  df-vd1 43633  df-vd2 43641
This theorem is referenced by: (None)
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