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Theorem vk15.4j 40847
Description: Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 40847 is vk15.4jVD 41233 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
vk15.4j.1 ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
vk15.4j.2 (∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏))
vk15.4j.3 ¬ ∀𝑥(𝜏𝜑)
Assertion
Ref Expression
vk15.4j (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Proof of Theorem vk15.4j
StepHypRef Expression
1 vk15.4j.3 . . . . . 6 ¬ ∀𝑥(𝜏𝜑)
2 exanali 1852 . . . . . 6 (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏𝜑))
31, 2mpbir 233 . . . . 5 𝑥(𝜏 ∧ ¬ 𝜑)
4 vk15.4j.2 . . . . . 6 (∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏))
5 alex 1819 . . . . . . . . . 10 (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
65biimpri 230 . . . . . . . . 9 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
7619.21bi 2180 . . . . . . . 8 (¬ ∃𝑥 ¬ 𝜃𝜃)
8 simpl 485 . . . . . . . . 9 ((𝜏 ∧ ¬ 𝜑) → 𝜏)
98a1i 11 . . . . . . . 8 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → 𝜏))
10 19.8a 2172 . . . . . . . 8 ((𝜃𝜏) → ∃𝑥(𝜃𝜏))
117, 9, 10syl6an 682 . . . . . . 7 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥(𝜃𝜏)))
12 notnot 144 . . . . . . 7 (∃𝑥(𝜃𝜏) → ¬ ¬ ∃𝑥(𝜃𝜏))
1311, 12syl6 35 . . . . . 6 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ ¬ ∃𝑥(𝜃𝜏)))
14 con3 156 . . . . . 6 ((∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏)) → (¬ ¬ ∃𝑥(𝜃𝜏) → ¬ ∀𝑥𝜒))
154, 13, 14mpsylsyld 69 . . . . 5 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ ∀𝑥𝜒))
16 hbe1 2140 . . . . . 6 (∃𝑥 ¬ 𝜃 → ∀𝑥𝑥 ¬ 𝜃)
1716hbn 2296 . . . . 5 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
18 hbn1 2139 . . . . 5 (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
193, 15, 17, 18eexinst01 40845 . . . 4 (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜒)
20 exnal 1820 . . . 4 (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
2119, 20sylibr 236 . . 3 (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜒)
22 vk15.4j.1 . . . . . . . . 9 ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
23 pm3.13 990 . . . . . . . . 9 (¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
2422, 23ax-mp 5 . . . . . . . 8 (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
25 simpr 487 . . . . . . . . . . . 12 ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
2625a1i 11 . . . . . . . . . . 11 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑))
27 19.8a 2172 . . . . . . . . . . 11 𝜑 → ∃𝑥 ¬ 𝜑)
2826, 27syl6 35 . . . . . . . . . 10 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥 ¬ 𝜑))
29 hbe1 2140 . . . . . . . . . 10 (∃𝑥 ¬ 𝜑 → ∀𝑥𝑥 ¬ 𝜑)
303, 28, 17, 29eexinst01 40845 . . . . . . . . 9 (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜑)
31 notnot 144 . . . . . . . . 9 (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
3230, 31syl 17 . . . . . . . 8 (¬ ∃𝑥 ¬ 𝜃 → ¬ ¬ ∃𝑥 ¬ 𝜑)
33 pm2.53 847 . . . . . . . 8 ((¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ¬ ∃𝑥 ¬ 𝜑 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
3424, 32, 33mpsyl 68 . . . . . . 7 (¬ ∃𝑥 ¬ 𝜃 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35 exanali 1852 . . . . . . . 8 (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓𝜒))
3635con5i 40842 . . . . . . 7 (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓𝜒))
3734, 36syl 17 . . . . . 6 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥(𝜓𝜒))
383719.21bi 2180 . . . . 5 (¬ ∃𝑥 ¬ 𝜃 → (𝜓𝜒))
3938con3d 155 . . . 4 (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ¬ 𝜓))
40 19.8a 2172 . . . 4 𝜓 → ∃𝑥 ¬ 𝜓)
4139, 40syl6 35 . . 3 (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ∃𝑥 ¬ 𝜓))
42 hbe1 2140 . . 3 (∃𝑥 ¬ 𝜓 → ∀𝑥𝑥 ¬ 𝜓)
4321, 41, 17, 42eexinst11 40846 . 2 (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜓)
44 exnal 1820 . 2 (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
4543, 44sylib 220 1 (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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