Theorem vk15.4j 45068

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 45068 is vk15.4jVD 45453 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 1878	. . . . . 6 ⊢ (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏 → 𝜑))
3	1, 2	mpbir 233	. . . . 5 ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
4		vk15.4j.2	. . . . . 6 ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏))
5		alex 1845	. . . . . . . . . 10 ⊢ (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
6	5	biimpri 230	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
7	6	19.21bi 2223	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → 𝜃)
8		simpl 486	. . . . . . . . 9 ⊢ ((𝜏 ∧ ¬ 𝜑) → 𝜏)
9	8	a1i 11	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → 𝜏))
10		19.8a 2215	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏) → ∃𝑥(𝜃 ∧ 𝜏))
11	7, 9, 10	syl6an 694	. . . . . . 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 2176	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜃 → ∀𝑥∃𝑥 ¬ 𝜃)
17	16	hbn 2328	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
18		hbn1 2175	. . . . 5 ⊢ (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
19	3, 15, 17, 18	eexinst01 45066	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜒)
20		exnal 1846	. . . 4 ⊢ (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
21	19, 20	sylibr 236	. . 3 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜒)
22		vk15.4j.1	. . . . . . . . 9 ⊢ ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
23		pm3.13 1007	. . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
26	25	a1i 11	. . . . . . . . . . 11 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑))
27		19.8a 2215	. . . . . . . . . . 11 ⊢ (¬ 𝜑 → ∃𝑥 ¬ 𝜑)
28	26, 27	syl6 35	. . . . . . . . . 10 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥 ¬ 𝜑))
29		hbe1 2176	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
30	3, 28, 17, 29	eexinst01 45066	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜑)
31		notnot 142	. . . . . . . . 9 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
32	30, 31	syl 17	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ¬ ∃𝑥 ¬ 𝜑)
33		pm2.53 862	. . . . . . . 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 1878	. . . . . . . 8 ⊢ (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓 → 𝜒))
36	35	con5i 45063	. . . . . . 7 ⊢ (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓 → 𝜒))
37	34, 36	syl 17	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥(𝜓 → 𝜒))
38	37	19.21bi 2223	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (𝜓 → 𝜒))
39	38	con3d 152	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ¬ 𝜓))
40		19.8a 2215	. . . 4 ⊢ (¬ 𝜓 → ∃𝑥 ¬ 𝜓)
41	39, 40	syl6 35	. . 3 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ∃𝑥 ¬ 𝜓))
42		hbe1 2176	. . 3 ⊢ (∃𝑥 ¬ 𝜓 → ∀𝑥∃𝑥 ¬ 𝜓)
43	21, 41, 17, 42	eexinst11 45067	. 2 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜓)
44		exnal 1846	. 2 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
45	43, 44	sylib 220	1 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1799  df-nf 1803

This theorem is referenced by: (None)