Theorem sbgoldbalt 47795

Description: An alternate (related to the original) formulation of the binary Goldbach conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. (Contributed by AV, 22-Jul-2020.)

Assertion

Ref	Expression
sbgoldbalt	⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Distinct variable group:   𝑛,𝑝,𝑞

Proof of Theorem sbgoldbalt

Step	Hyp	Ref	Expression
1		2z 12624	. . . . . 6 ⊢ 2 ∈ ℤ
2		evenz 47644	. . . . . 6 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3		zltp1le 12642	. . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
4	1, 2, 3	sylancr 587	. . . . 5 ⊢ (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5		2p1e3 12382	. . . . . . 7 ⊢ (2 + 1) = 3
6	5	breq1i 5126	. . . . . 6 ⊢ ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7		3re 12320	. . . . . . . . 9 ⊢ 3 ∈ ℝ
8	7	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ Even → 3 ∈ ℝ)
9	2	zred 12697	. . . . . . . 8 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
10	8, 9	leloed 11378	. . . . . . 7 ⊢ (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11		3z 12625	. . . . . . . . . . . 12 ⊢ 3 ∈ ℤ
12		zltp1le 12642	. . . . . . . . . . . 12 ⊢ ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
13	11, 2, 12	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14		3p1e4 12385	. . . . . . . . . . . . 13 ⊢ (3 + 1) = 4
15	14	breq1i 5126	. . . . . . . . . . . 12 ⊢ ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16		4re 12324	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ
17	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ Even → 4 ∈ ℝ)
18	17, 9	leloed 11378	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19		pm3.35 802	. . . . . . . . . . . . . . . . . 18 ⊢ ((4 < 𝑛 ∧ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20		isgbe 47765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21		simp3 1138	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
22	21	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞)))
23	22	reximdva 3153	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
24	23	reximdva 3153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ Even → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
25	24	imp 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
26	20, 25	sylbi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ GoldbachEven → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
27	26	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ GoldbachEven → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
28	19, 27	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((4 < 𝑛 ∧ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )) → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
29	28	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (4 < 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
30	29	com23 86	. . . . . . . . . . . . . . 15 ⊢ (4 < 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
31		2prm 16711	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℙ
32		2p2e4 12375	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 + 2) = 4
33	32	eqcomi 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 = (2 + 2)
34		rspceov 7454	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
35	31, 31, 33, 34	mp3an 1463	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36		eqeq1 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37	36	2rexbidv 3206	. . . . . . . . . . . . . . . . . 18 ⊢ (4 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
38	35, 37	mpbii 233	. . . . . . . . . . . . . . . . 17 ⊢ (4 = 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
39	38	a1d 25	. . . . . . . . . . . . . . . 16 ⊢ (4 = 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
40	39	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (4 = 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
41	30, 40	jaoi 857	. . . . . . . . . . . . . 14 ⊢ ((4 < 𝑛 ∨ 4 = 𝑛) → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
42	41	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ Even → ((4 < 𝑛 ∨ 4 = 𝑛) → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
43	18, 42	sylbid 240	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ Even → (4 ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
44	15, 43	biimtrid 242	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → ((3 + 1) ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
45	13, 44	sylbid 240	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → (3 < 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
46	45	com12 32	. . . . . . . . 9 ⊢ (3 < 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
47		3odd 47722	. . . . . . . . . . . 12 ⊢ 3 ∈ Odd
48		eleq1 2822	. . . . . . . . . . . 12 ⊢ (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
49	47, 48	mpbii 233	. . . . . . . . . . 11 ⊢ (3 = 𝑛 → 𝑛 ∈ Odd )
50		oddneven 47658	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Odd → ¬ 𝑛 ∈ Even )
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ (3 = 𝑛 → ¬ 𝑛 ∈ Even )
52	51	pm2.21d 121	. . . . . . . . 9 ⊢ (3 = 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
53	46, 52	jaoi 857	. . . . . . . 8 ⊢ ((3 < 𝑛 ∨ 3 = 𝑛) → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
54	53	com12 32	. . . . . . 7 ⊢ (𝑛 ∈ Even → ((3 < 𝑛 ∨ 3 = 𝑛) → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
55	10, 54	sylbid 240	. . . . . 6 ⊢ (𝑛 ∈ Even → (3 ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
56	6, 55	biimtrid 242	. . . . 5 ⊢ (𝑛 ∈ Even → ((2 + 1) ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
57	4, 56	sylbid 240	. . . 4 ⊢ (𝑛 ∈ Even → (2 < 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
58	57	com23 86	. . 3 ⊢ (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
59		2lt4 12415	. . . . . . . 8 ⊢ 2 < 4
60		2re 12314	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
61	60	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → 2 ∈ ℝ)
62		lttr 11311	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
63	61, 17, 9, 62	syl3anc 1373	. . . . . . . 8 ⊢ (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
64	59, 63	mpani 696	. . . . . . 7 ⊢ (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
65	64	imp 406	. . . . . 6 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66		simpll 766	. . . . . . . . 9 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ Even )
67		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
68	67	anim1i 615	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
69	68	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
70		simpll 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 ∈ Even ∧ 4 < 𝑛))
71	70	anim1i 615	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
72		df-3an 1088	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
73	71, 72	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 = (𝑝 + 𝑞)))
74		sbgoldbaltlem2 47794	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd )))
75	69, 73, 74	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ))
76		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
77		df-3an 1088	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) ∧ 𝑛 = (𝑝 + 𝑞)))
78	75, 76, 77	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
79	78	ex 412	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 = (𝑝 + 𝑞) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
80	79	reximdva 3153	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
81	80	reximdva 3153	. . . . . . . . . 10 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
82	81	imp 406	. . . . . . . . 9 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
83	66, 82	jca 511	. . . . . . . 8 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
84	83	ex 412	. . . . . . 7 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))))
85	84, 20	imbitrrdi 252	. . . . . 6 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → 𝑛 ∈ GoldbachEven ))
86	65, 85	embantd 59	. . . . 5 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ GoldbachEven ))
87	86	ex 412	. . . 4 ⊢ (𝑛 ∈ Even → (4 < 𝑛 → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ GoldbachEven )))
88	87	com23 86	. . 3 ⊢ (𝑛 ∈ Even → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
89	58, 88	impbid 212	. 2 ⊢ (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
90	89	ralbiia 3080	1 ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   class class class wbr 5119  (class class class)co 7405  ℝcr 11128  1c1 11130   + caddc 11132   < clt 11269   ≤ cle 11270  2c2 12295  3c3 12296  4c4 12297  ℤcz 12588  ℙcprime 16690   Even ceven 47638   Odd codd 47639   GoldbachEven cgbe 47759

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-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-n0 12502  df-z 12589  df-uz 12853  df-rp 13009  df-fz 13525  df-seq 14020  df-exp 14080  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-dvds 16273  df-prm 16691  df-even 47640  df-odd 47641  df-gbe 47762

This theorem is referenced by:  sbgoldbb  47796  sbgoldbmb  47800