Theorem vk15.4j 39430

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 39430 is vk15.4jVD 39826 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 1955	. . . . . 6 ⊢ (∃𝑥(𝜏 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜏 → 𝜑))
3	1, 2	mpbir 222	. . . . 5 ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
4		vk15.4j.2	. . . . . 6 ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏))
5		alex 1920	. . . . . . . . . 10 ⊢ (∀𝑥𝜃 ↔ ¬ ∃𝑥 ¬ 𝜃)
6	5	biimpri 219	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥𝜃)
7	6	19.21bi 2221	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → 𝜃)
8		simpl 474	. . . . . . . . 9 ⊢ ((𝜏 ∧ ¬ 𝜑) → 𝜏)
9	8	a1i 11	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → 𝜏))
10		19.8a 2214	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏) → ∃𝑥(𝜃 ∧ 𝜏))
11	7, 9, 10	syl6an 674	. . . . . . 7 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥(𝜃 ∧ 𝜏)))
12		notnot 138	. . . . . . 7 ⊢ (∃𝑥(𝜃 ∧ 𝜏) → ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏))
13	11, 12	syl6 35	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏)))
14		con3 150	. . . . . 6 ⊢ ((∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏)) → (¬ ¬ ∃𝑥(𝜃 ∧ 𝜏) → ¬ ∀𝑥𝜒))
15	4, 13, 14	mpsylsyld 69	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ ∀𝑥𝜒))
16		hbe1 2185	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜃 → ∀𝑥∃𝑥 ¬ 𝜃)
17	16	hbn 2321	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥 ¬ ∃𝑥 ¬ 𝜃)
18		hbn1 2184	. . . . 5 ⊢ (¬ ∀𝑥𝜒 → ∀𝑥 ¬ ∀𝑥𝜒)
19	3, 15, 17, 18	eexinst01 39428	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜒)
20		exnal 1921	. . . 4 ⊢ (∃𝑥 ¬ 𝜒 ↔ ¬ ∀𝑥𝜒)
21	19, 20	sylibr 225	. . 3 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜒)
22		vk15.4j.1	. . . . . . . . 9 ⊢ ¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))
23		pm3.13 1017	. . . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑)
26	25	a1i 11	. . . . . . . . . . 11 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ¬ 𝜑))
27		19.8a 2214	. . . . . . . . . . 11 ⊢ (¬ 𝜑 → ∃𝑥 ¬ 𝜑)
28	26, 27	syl6 35	. . . . . . . . . 10 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ((𝜏 ∧ ¬ 𝜑) → ∃𝑥 ¬ 𝜑))
29		hbe1 2185	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
30	3, 28, 17, 29	eexinst01 39428	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜑)
31		notnot 138	. . . . . . . . 9 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ¬ ∃𝑥 ¬ 𝜑)
32	30, 31	syl 17	. . . . . . . 8 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ¬ ∃𝑥 ¬ 𝜑)
33		pm2.53 877	. . . . . . . 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 1955	. . . . . . . 8 ⊢ (∃𝑥(𝜓 ∧ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜓 → 𝜒))
36	35	con5i 39425	. . . . . . 7 ⊢ (¬ ∃𝑥(𝜓 ∧ ¬ 𝜒) → ∀𝑥(𝜓 → 𝜒))
37	34, 36	syl 17	. . . . . 6 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∀𝑥(𝜓 → 𝜒))
38	37	19.21bi 2221	. . . . 5 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (𝜓 → 𝜒))
39	38	con3d 149	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ¬ 𝜓))
40		19.8a 2214	. . . 4 ⊢ (¬ 𝜓 → ∃𝑥 ¬ 𝜓)
41	39, 40	syl6 35	. . 3 ⊢ (¬ ∃𝑥 ¬ 𝜃 → (¬ 𝜒 → ∃𝑥 ¬ 𝜓))
42		hbe1 2185	. . 3 ⊢ (∃𝑥 ¬ 𝜓 → ∀𝑥∃𝑥 ¬ 𝜓)
43	21, 41, 17, 42	eexinst11 39429	. 2 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ∃𝑥 ¬ 𝜓)
44		exnal 1921	. 2 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
45	43, 44	sylib 209	1 ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∨ wo 873  ∀wal 1650  ∃wex 1874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-12 2211

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879

This theorem is referenced by: (None)