Theorem sbgoldbalt 49018

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 12738	. . . . . 6 ⊢ 2 ∈ ℤ
2		evenz 48867	. . . . . 6 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3		zltp1le 12756	. . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
4	1, 2, 3	sylancr 599	. . . . 5 ⊢ (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5		2p1e3 12494	. . . . . . 7 ⊢ (2 + 1) = 3
6	5	breq1i 5112	. . . . . 6 ⊢ ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7		3re 12433	. . . . . . . . 9 ⊢ 3 ∈ ℝ
8	7	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ Even → 3 ∈ ℝ)
9	2	zred 12813	. . . . . . . 8 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
10	8, 9	leloed 11463	. . . . . . 7 ⊢ (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11		3z 12739	. . . . . . . . . . . 12 ⊢ 3 ∈ ℤ
12		zltp1le 12756	. . . . . . . . . . . 12 ⊢ ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
13	11, 2, 12	sylancr 599	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14		3p1e4 12497	. . . . . . . . . . . . 13 ⊢ (3 + 1) = 4
15	14	breq1i 5112	. . . . . . . . . . . 12 ⊢ ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16		4re 12437	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ
17	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ Even → 4 ∈ ℝ)
18	17, 9	leloed 11463	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19		pm3.35 815	. . . . . . . . . . . . . . . . . 18 ⊢ ((4 < 𝑛 ∧ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20		isgbe 48988	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21		simp3 1156	. . . . . . . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
24	23	reximdva 3178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ Even → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
25	24	imp 412	. . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . 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 16879	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℙ
32		2p2e4 12487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 + 2) = 4
33	32	eqcomi 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 = (2 + 2)
34		rspceov 7476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
35	31, 31, 33, 34	mp3an 1490	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36		eqeq1 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37	36	2rexbidv 3230	. . . . . . . . . . . . . . . . . 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 871	. . . . . . . . . . . . . 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 48945	. . . . . . . . . . . 12 ⊢ 3 ∈ Odd
48		eleq1 2851	. . . . . . . . . . . 12 ⊢ (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
49	47, 48	mpbii 236	. . . . . . . . . . 11 ⊢ (3 = 𝑛 → 𝑛 ∈ Odd )
50		oddneven 48881	. . . . . . . . . . 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 871	. . . . . . . 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 12530	. . . . . . . 8 ⊢ 2 < 4
60		2re 12427	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
61	60	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → 2 ∈ ℝ)
62		lttr 11396	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
63	61, 17, 9, 62	syl3anc 1398	. . . . . . . 8 ⊢ (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
64	59, 63	mpani 709	. . . . . . 7 ⊢ (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
65	64	imp 412	. . . . . 6 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66		simpll 779	. . . . . . . . 9 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ Even )
67		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
68	67	anim1i 627	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
69	68	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
70		simpll 779	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 ∈ Even ∧ 4 < 𝑛))
71	70	anim1i 627	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
72		df-3an 1105	. . . . . . . . . . . . . . . 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 49017	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 14 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
77		df-3an 1105	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) ∧ 𝑛 = (𝑝 + 𝑞)))
78	75, 76, 77	sylanbrc 595	. . . . . . . . . . . . 13 ⊢ (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
79	78	ex 418	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 = (𝑝 + 𝑞) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
80	79	reximdva 3178	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
81	80	reximdva 3178	. . . . . . . . . 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 412	. . . . . . . . 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 521	. . . . . . . 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 418	. . . . . . 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 418	. . . 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 3109	1 ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3079  ∃wrex 3089   class class class wbr 5105  (class class class)co 7427  ℝcr 11209  1c1 11211   + caddc 11213   < clt 11353   ≤ cle 11354  2c2 12407  3c3 12408  4c4 12409  ℤcz 12703  ℙcprime 16858   Even ceven 48861   Odd codd 48862   GoldbachEven cgbe 48982

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287  ax-pre-sup 11288

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  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 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-2o 8488  df-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9444  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-div 11984  df-nn 12346  df-2 12415  df-3 12416  df-4 12417  df-n0 12617  df-z 12704  df-uz 12976  df-rp 13131  df-fz 13650  df-seq 14155  df-exp 14215  df-cj 15278  df-re 15279  df-im 15280  df-sqrt 15414  df-abs 15415  df-dvds 16435  df-prm 16859  df-even 48863  df-odd 48864  df-gbe 48985

This theorem is used by:  sbgoldbb  49019  sbgoldbmb  49023