Theorem vk15.4j 44548

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 44548 is vk15.4jVD 44934 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 1859	. . . . . 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 1826	. . . . . . . . . 10 ⊢ (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
6	5	biimpri 228	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
7	6	19.21bi 2189	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → 𝜃)
8		simpl 482	. . . . . . . . 9 ⊢ ((𝜏 ∧ ¬ 𝜑) → 𝜏)
9	8	a1i 11	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → 𝜏))
10		19.8a 2181	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏) → ∃𝑥(𝜃 ∧ 𝜏))
11	7, 9, 10	syl6an 684	. . . . . . 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 2143	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜃 → ∀𝑥∃𝑥 ¬ 𝜃)
17	16	hbn 2295	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
18		hbn1 2142	. . . . 5 ⊢ (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
19	3, 15, 17, 18	eexinst01 44546	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜒)
20		exnal 1827	. . . 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 2181	. . . . . . . . . . 11 ⊢ (¬ 𝜑 → ∃𝑥 ¬ 𝜑)
28	26, 27	syl6 35	. . . . . . . . . 10 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥 ¬ 𝜑))
29		hbe1 2143	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
30	3, 28, 17, 29	eexinst01 44546	. . . . . . . . 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 1859	. . . . . . . 8 ⊢ (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓 → 𝜒))
36	35	con5i 44543	. . . . . . 7 ⊢ (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓 → 𝜒))
37	34, 36	syl 17	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥(𝜓 → 𝜒))
38	37	19.21bi 2189	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (𝜓 → 𝜒))
39	38	con3d 152	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ¬ 𝜓))
40		19.8a 2181	. . . 4 ⊢ (¬ 𝜓 → ∃𝑥 ¬ 𝜓)
41	39, 40	syl6 35	. . 3 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ∃𝑥 ¬ 𝜓))
42		hbe1 2143	. . 3 ⊢ (∃𝑥 ¬ 𝜓 → ∀𝑥∃𝑥 ¬ 𝜓)
43	21, 41, 17, 42	eexinst11 44547	. 2 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜓)
44		exnal 1827	. 2 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
45	43, 44	sylib 218	1 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1538  ∃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 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)