Theorem sbgoldbalt 48495

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 12629	. . . . . 6 ⊢ 2 ∈ ℤ
2		evenz 48344	. . . . . 6 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3		zltp1le 12647	. . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
4	1, 2, 3	sylancr 598	. . . . 5 ⊢ (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5		2p1e3 12385	. . . . . . 7 ⊢ (2 + 1) = 3
6	5	breq1i 5121	. . . . . 6 ⊢ ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7		3re 12324	. . . . . . . . 9 ⊢ 3 ∈ ℝ
8	7	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ Even → 3 ∈ ℝ)
9	2	zred 12703	. . . . . . . 8 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
10	8, 9	leloed 11356	. . . . . . 7 ⊢ (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11		3z 12630	. . . . . . . . . . . 12 ⊢ 3 ∈ ℤ
12		zltp1le 12647	. . . . . . . . . . . 12 ⊢ ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
13	11, 2, 12	sylancr 598	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14		3p1e4 12388	. . . . . . . . . . . . 13 ⊢ (3 + 1) = 4
15	14	breq1i 5121	. . . . . . . . . . . 12 ⊢ ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16		4re 12328	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ
17	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ Even → 4 ∈ ℝ)
18	17, 9	leloed 11356	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19		pm3.35 814	. . . . . . . . . . . . . . . . . 18 ⊢ ((4 < 𝑛 ∧ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20		isgbe 48465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21		simp3 1154	. . . . . . . . . . . . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
24	23	reximdva 3185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ Even → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
25	24	imp 411	. . . . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ GoldbachEven → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
27	26	a1d 26	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ GoldbachEven → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
28	19, 27	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((4 < 𝑛 ∧ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )) → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
29	28	ex 417	. . . . . . . . . . . . . . . 16 ⊢ (4 < 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
30	29	com23 87	. . . . . . . . . . . . . . 15 ⊢ (4 < 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
31		2prm 16753	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℙ
32		2p2e4 12378	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 + 2) = 4
33	32	eqcomi 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 = (2 + 2)
34		rspceov 7463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
35	31, 31, 33, 34	mp3an 1488	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36		eqeq1 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37	36	2rexbidv 3237	. . . . . . . . . . . . . . . . . 18 ⊢ (4 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
38	35, 37	mpbii 236	. . . . . . . . . . . . . . . . 17 ⊢ (4 = 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
39	38	a1d 26	. . . . . . . . . . . . . . . 16 ⊢ (4 = 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
40	39	a1d 26	. . . . . . . . . . . . . . 15 ⊢ (4 = 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
41	30, 40	jaoi 870	. . . . . . . . . . . . . 14 ⊢ ((4 < 𝑛 ∨ 4 = 𝑛) → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
42	41	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ Even → ((4 < 𝑛 ∨ 4 = 𝑛) → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
43	18, 42	sylbid 243	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ Even → (4 ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
44	15, 43	biimtrid 245	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → ((3 + 1) ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
45	13, 44	sylbid 243	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → (3 < 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
46	45	com12 33	. . . . . . . . 9 ⊢ (3 < 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
47		3odd 48422	. . . . . . . . . . . 12 ⊢ 3 ∈ Odd
48		eleq1 2858	. . . . . . . . . . . 12 ⊢ (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
49	47, 48	mpbii 236	. . . . . . . . . . 11 ⊢ (3 = 𝑛 → 𝑛 ∈ Odd )
50		oddneven 48358	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Odd → ¬ 𝑛 ∈ Even )
51	49, 50	syl 18	. . . . . . . . . 10 ⊢ (3 = 𝑛 → ¬ 𝑛 ∈ Even )
52	51	pm2.21d 122	. . . . . . . . 9 ⊢ (3 = 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
53	46, 52	jaoi 870	. . . . . . . 8 ⊢ ((3 < 𝑛 ∨ 3 = 𝑛) → (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
54	53	com12 33	. . . . . . 7 ⊢ (𝑛 ∈ Even → ((3 < 𝑛 ∨ 3 = 𝑛) → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
55	10, 54	sylbid 243	. . . . . 6 ⊢ (𝑛 ∈ Even → (3 ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
56	6, 55	biimtrid 245	. . . . 5 ⊢ (𝑛 ∈ Even → ((2 + 1) ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
57	4, 56	sylbid 243	. . . 4 ⊢ (𝑛 ∈ Even → (2 < 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
58	57	com23 87	. . 3 ⊢ (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
59		2lt4 12421	. . . . . . . 8 ⊢ 2 < 4
60		2re 12318	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
61	60	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → 2 ∈ ℝ)
62		lttr 11289	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
63	61, 17, 9, 62	syl3anc 1396	. . . . . . . 8 ⊢ (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
64	59, 63	mpani 708	. . . . . . 7 ⊢ (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
65	64	imp 411	. . . . . 6 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66		simpll 778	. . . . . . . . 9 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ Even )
67		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
68	67	anim1i 626	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
69	68	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
70		simpll 778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 ∈ Even ∧ 4 < 𝑛))
71	70	anim1i 626	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
72		df-3an 1103	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
73	71, 72	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 = (𝑝 + 𝑞)))
74		sbgoldbaltlem2 48494	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd )))
75	69, 73, 74	sylc 66	. . . . . . . . . . . . . 14 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ))
76		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
77		df-3an 1103	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) ∧ 𝑛 = (𝑝 + 𝑞)))
78	75, 76, 77	sylanbrc 594	. . . . . . . . . . . . 13 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
79	78	ex 417	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 = (𝑝 + 𝑞) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
80	79	reximdva 3185	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
81	80	reximdva 3185	. . . . . . . . . 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 411	. . . . . . . . 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 520	. . . . . . . 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 417	. . . . . . 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 255	. . . . . 6 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → 𝑛 ∈ GoldbachEven ))
86	65, 85	embantd 60	. . . . 5 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ GoldbachEven ))
87	86	ex 417	. . . 4 ⊢ (𝑛 ∈ Even → (4 < 𝑛 → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ GoldbachEven )))
88	87	com23 87	. . 3 ⊢ (𝑛 ∈ Even → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
89	58, 88	impbid 215	. 2 ⊢ (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
90	89	ralbiia 3116	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 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ∀wral 3086  ∃wrex 3096   class class class wbr 5114  (class class class)co 7414  ℝcr 11102  1c1 11104   + caddc 11106   < clt 11246   ≤ cle 11247  2c2 12298  3c3 12299  4c4 12300  ℤcz 12594  ℙcprime 16732   Even ceven 48338   Odd codd 48339   GoldbachEven cgbe 48459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-cnex 11159  ax-resscn 11160  ax-1cn 11161  ax-icn 11162  ax-addcl 11163  ax-addrcl 11164  ax-mulcl 11165  ax-mulrcl 11166  ax-mulcom 11167  ax-addass 11168  ax-mulass 11169  ax-distr 11170  ax-i2m1 11171  ax-1ne0 11172  ax-1rid 11173  ax-rnegex 11174  ax-rrecex 11175  ax-cnre 11176  ax-pre-lttri 11177  ax-pre-lttrn 11178  ax-pre-ltadd 11179  ax-pre-mulgt0 11180  ax-pre-sup 11181

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-2o 8457  df-er 8697  df-en 8947  df-dom 8948  df-sdom 8949  df-fin 8950  df-sup 9405  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11446  df-neg 11447  df-div 11875  df-nn 12237  df-2 12306  df-3 12307  df-4 12308  df-n0 12508  df-z 12595  df-uz 12866  df-rp 13020  df-fz 13539  df-seq 14041  df-exp 14101  df-cj 15153  df-re 15154  df-im 15155  df-sqrt 15289  df-abs 15290  df-dvds 16314  df-prm 16733  df-even 48340  df-odd 48341  df-gbe 48462

This theorem is referenced by:  sbgoldbb  48496  sbgoldbmb  48500