Theorem vk15.4j 44878

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 44878 is vk15.4jVD 45263 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(𝜃 ∧ 𝜏))
vk15.4j.3	⊢ ¬ ∀𝑥(𝜏 → 𝜑)

Assertion

Ref	Expression
vk15.4j	⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Proof of Theorem vk15.4j

Step	Hyp	Ref	Expression
1		vk15.4j.3	. . . . . 6 ⊢ ¬ ∀𝑥(𝜏 → 𝜑)
2		exanali 1861	. . . . . 6 ⊢ (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏 → 𝜑))
3	1, 2	mpbir 231	. . . . 5 ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
4		vk15.4j.2	. . . . . 6 ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏))
5		alex 1828	. . . . . . . . . 10 ⊢ (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
6	5	biimpri 228	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
7	6	19.21bi 2197	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → 𝜃)
8		simpl 482	. . . . . . . . 9 ⊢ ((𝜏 ∧ ¬ 𝜑) → 𝜏)
9	8	a1i 11	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → 𝜏))
10		19.8a 2189	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏) → ∃𝑥(𝜃 ∧ 𝜏))
11	7, 9, 10	syl6an 685	. . . . . . 7 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥(𝜃 ∧ 𝜏)))
12		notnot 142	. . . . . . 7 ⊢ (∃𝑥(𝜃 ∧ 𝜏) → ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏))
13	11, 12	syl6 35	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏)))
14		con3 153	. . . . . 6 ⊢ ((∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏)) → (¬ ¬ ∃𝑥(𝜃 ∧ 𝜏) → ¬ ∀𝑥𝜒))
15	4, 13, 14	mpsylsyld 69	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ ∀𝑥𝜒))
16		hbe1 2149	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜃 → ∀𝑥∃𝑥 ¬ 𝜃)
17	16	hbn 2302	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
18		hbn1 2148	. . . . 5 ⊢ (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
19	3, 15, 17, 18	eexinst01 44876	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜒)
20		exnal 1829	. . . 4 ⊢ (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
21	19, 20	sylibr 234	. . 3 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜒)
22		vk15.4j.1	. . . . . . . . 9 ⊢ ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
23		pm3.13 997	. . . . . . . . 9 ⊢ (¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
24	22, 23	ax-mp 5	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
25		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
26	25	a1i 11	. . . . . . . . . . 11 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑))
27		19.8a 2189	. . . . . . . . . . 11 ⊢ (¬ 𝜑 → ∃𝑥 ¬ 𝜑)
28	26, 27	syl6 35	. . . . . . . . . 10 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥 ¬ 𝜑))
29		hbe1 2149	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
30	3, 28, 17, 29	eexinst01 44876	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜑)
31		notnot 142	. . . . . . . . 9 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
32	30, 31	syl 17	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ¬ ∃𝑥 ¬ 𝜑)
33		pm2.53 852	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) → (¬ ¬ ∃𝑥 ¬ 𝜑 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)))
34	24, 32, 33	mpsyl 68	. . . . . . 7 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒))
35		exanali 1861	. . . . . . . 8 ⊢ (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓 → 𝜒))
36	35	con5i 44873	. . . . . . 7 ⊢ (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓 → 𝜒))
37	34, 36	syl 17	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥(𝜓 → 𝜒))
38	37	19.21bi 2197	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (𝜓 → 𝜒))
39	38	con3d 152	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ¬ 𝜓))
40		19.8a 2189	. . . 4 ⊢ (¬ 𝜓 → ∃𝑥 ¬ 𝜓)
41	39, 40	syl6 35	. . 3 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ∃𝑥 ¬ 𝜓))
42		hbe1 2149	. . 3 ⊢ (∃𝑥 ¬ 𝜓 → ∀𝑥∃𝑥 ¬ 𝜓)
43	21, 41, 17, 42	eexinst11 44877	. 2 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜓)
44		exnal 1829	. 2 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
45	43, 44	sylib 218	1 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1540  ∃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 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)