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Theorem vk15.4j 44526
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 44526 is vk15.4jVD 44912 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 1857 . . . . . 6 (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏𝜑))
31, 2mpbir 231 . . . . 5 𝑥(𝜏 ∧ ¬ 𝜑)
4 vk15.4j.2 . . . . . 6 (∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏))
5 alex 1823 . . . . . . . . . 10 (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
65biimpri 228 . . . . . . . . 9 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
7619.21bi 2187 . . . . . . . 8 (¬ ∃𝑥 ¬ 𝜃𝜃)
8 simpl 482 . . . . . . . . 9 ((𝜏 ∧ ¬ 𝜑) → 𝜏)
98a1i 11 . . . . . . . 8 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → 𝜏))
10 19.8a 2179 . . . . . . . 8 ((𝜃𝜏) → ∃𝑥(𝜃𝜏))
117, 9, 10syl6an 684 . . . . . . 7 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥(𝜃𝜏)))
12 notnot 142 . . . . . . 7 (∃𝑥(𝜃𝜏) → ¬ ¬ ∃𝑥(𝜃𝜏))
1311, 12syl6 35 . . . . . 6 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ ¬ ∃𝑥(𝜃𝜏)))
14 con3 153 . . . . . 6 ((∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏)) → (¬ ¬ ∃𝑥(𝜃𝜏) → ¬ ∀𝑥𝜒))
154, 13, 14mpsylsyld 69 . . . . 5 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ ∀𝑥𝜒))
16 hbe1 2141 . . . . . 6 (∃𝑥 ¬ 𝜃 → ∀𝑥𝑥 ¬ 𝜃)
1716hbn 2294 . . . . 5 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
18 hbn1 2140 . . . . 5 (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
193, 15, 17, 18eexinst01 44524 . . . 4 (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜒)
20 exnal 1824 . . . 4 (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
2119, 20sylibr 234 . . 3 (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜒)
22 vk15.4j.1 . . . . . . . . 9 ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
23 pm3.13 996 . . . . . . . . 9 (¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
2422, 23ax-mp 5 . . . . . . . 8 (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
25 simpr 484 . . . . . . . . . . . 12 ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
2625a1i 11 . . . . . . . . . . 11 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑))
27 19.8a 2179 . . . . . . . . . . 11 𝜑 → ∃𝑥 ¬ 𝜑)
2826, 27syl6 35 . . . . . . . . . 10 (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥 ¬ 𝜑))
29 hbe1 2141 . . . . . . . . . 10 (∃𝑥 ¬ 𝜑 → ∀𝑥𝑥 ¬ 𝜑)
303, 28, 17, 29eexinst01 44524 . . . . . . . . 9 (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜑)
31 notnot 142 . . . . . . . . 9 (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
3230, 31syl 17 . . . . . . . 8 (¬ ∃𝑥 ¬ 𝜃 → ¬ ¬ ∃𝑥 ¬ 𝜑)
33 pm2.53 851 . . . . . . . 8 ((¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ¬ ∃𝑥 ¬ 𝜑 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
3424, 32, 33mpsyl 68 . . . . . . 7 (¬ ∃𝑥 ¬ 𝜃 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35 exanali 1857 . . . . . . . 8 (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓𝜒))
3635con5i 44521 . . . . . . 7 (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓𝜒))
3734, 36syl 17 . . . . . 6 (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥(𝜓𝜒))
383719.21bi 2187 . . . . 5 (¬ ∃𝑥 ¬ 𝜃 → (𝜓𝜒))
3938con3d 152 . . . 4 (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ¬ 𝜓))
40 19.8a 2179 . . . 4 𝜓 → ∃𝑥 ¬ 𝜓)
4139, 40syl6 35 . . 3 (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ∃𝑥 ¬ 𝜓))
42 hbe1 2141 . . 3 (∃𝑥 ¬ 𝜓 → ∀𝑥𝑥 ¬ 𝜓)
4321, 41, 17, 42eexinst11 44525 . 2 (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜓)
44 exnal 1824 . 2 (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
4543, 44sylib 218 1 (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  wal 1535  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781
This theorem is referenced by: (None)
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