Theorem sbgoldbalt 48272

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 12550	. . . . . 6 ⊢ 2 ∈ ℤ
2		evenz 48121	. . . . . 6 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3		zltp1le 12568	. . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
4	1, 2, 3	sylancr 593	. . . . 5 ⊢ (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5		2p1e3 12309	. . . . . . 7 ⊢ (2 + 1) = 3
6	5	breq1i 5079	. . . . . 6 ⊢ ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7		3re 12252	. . . . . . . . 9 ⊢ 3 ∈ ℝ
8	7	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ Even → 3 ∈ ℝ)
9	2	zred 12624	. . . . . . . 8 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
10	8, 9	leloed 11280	. . . . . . 7 ⊢ (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11		3z 12551	. . . . . . . . . . . 12 ⊢ 3 ∈ ℤ
12		zltp1le 12568	. . . . . . . . . . . 12 ⊢ ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
13	11, 2, 12	sylancr 593	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14		3p1e4 12312	. . . . . . . . . . . . 13 ⊢ (3 + 1) = 4
15	14	breq1i 5079	. . . . . . . . . . . 12 ⊢ ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16		4re 12256	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ
17	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ Even → 4 ∈ ℝ)
18	17, 9	leloed 11280	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19		pm3.35 808	. . . . . . . . . . . . . . . . . 18 ⊢ ((4 < 𝑛 ∧ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20		isgbe 48242	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21		simp3 1144	. . . . . . . . . . . . . . . . . . . . . . . 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 3152	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
24	23	reximdva 3152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ Even → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
25	24	imp 407	. . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 16652	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℙ
32		2p2e4 12302	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 + 2) = 4
33	32	eqcomi 2748	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 = (2 + 2)
34		rspceov 7405	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
35	31, 31, 33, 34	mp3an 1469	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36		eqeq1 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37	36	2rexbidv 3204	. . . . . . . . . . . . . . . . . 18 ⊢ (4 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
38	35, 37	mpbii 234	. . . . . . . . . . . . . . . . 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 863	. . . . . . . . . . . . . 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 241	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ Even → (4 ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
44	15, 43	biimtrid 243	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → ((3 + 1) ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
45	13, 44	sylbid 241	. . . . . . . . . 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 48199	. . . . . . . . . . . 12 ⊢ 3 ∈ Odd
48		eleq1 2827	. . . . . . . . . . . 12 ⊢ (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
49	47, 48	mpbii 234	. . . . . . . . . . 11 ⊢ (3 = 𝑛 → 𝑛 ∈ Odd )
50		oddneven 48135	. . . . . . . . . . 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 863	. . . . . . . 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 241	. . . . . 6 ⊢ (𝑛 ∈ Even → (3 ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
56	6, 55	biimtrid 243	. . . . 5 ⊢ (𝑛 ∈ Even → ((2 + 1) ≤ 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
57	4, 56	sylbid 241	. . . 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 12342	. . . . . . . 8 ⊢ 2 < 4
60		2re 12246	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
61	60	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → 2 ∈ ℝ)
62		lttr 11213	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
63	61, 17, 9, 62	syl3anc 1379	. . . . . . . 8 ⊢ (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
64	59, 63	mpani 702	. . . . . . 7 ⊢ (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
65	64	imp 407	. . . . . 6 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66		simpll 772	. . . . . . . . 9 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ Even )
67		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
68	67	anim1i 621	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
69	68	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
70		simpll 772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 ∈ Even ∧ 4 < 𝑛))
71	70	anim1i 621	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
72		df-3an 1094	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
73	71, 72	sylibr 235	. . . . . . . . . . . . . . 15 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 = (𝑝 + 𝑞)))
74		sbgoldbaltlem2 48271	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
77		df-3an 1094	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) ∧ 𝑛 = (𝑝 + 𝑞)))
78	75, 76, 77	sylanbrc 589	. . . . . . . . . . . . 13 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
79	78	ex 413	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 = (𝑝 + 𝑞) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
80	79	reximdva 3152	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
81	80	reximdva 3152	. . . . . . . . . 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 407	. . . . . . . . 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 516	. . . . . . . 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 413	. . . . . . 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 253	. . . . . 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 413	. . . 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 213	. 2 ⊢ (𝑛 ∈ Even → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
90	89	ralbiia 3083	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 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   class class class wbr 5072  (class class class)co 7356  ℝcr 11028  1c1 11030   + caddc 11032   < clt 11170   ≤ cle 11171  2c2 12227  3c3 12228  4c4 12229  ℤcz 12515  ℙcprime 16631   Even ceven 48115   Odd codd 48116   GoldbachEven cgbe 48236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16213  df-prm 16632  df-even 48117  df-odd 48118  df-gbe 48239

This theorem is referenced by:  sbgoldbb  48273  sbgoldbmb  48277