Theorem fmla1 36121

Description: The valid Godel formulas of height 1 is the set of all formulas of the form (𝑎⊼_𝑔𝑏) and ∀_𝑔𝑘𝑎 with atoms 𝑎, 𝑏 of the form 𝑥 ∈ 𝑦. (Contributed by AV, 20-Sep-2023.)

Assertion

Ref	Expression
fmla1	⊢ (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))})

Distinct variable group:   𝑖,𝑗,𝑥,𝑘,𝑙

Proof of Theorem fmla1

Dummy variables 𝑦 𝑛 𝑧 𝑚 𝑜 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-1o 8460	. . 3 ⊢ 1o = suc ∅
2	1	fveq2i 6880	. 2 ⊢ (Fmla‘1o) = (Fmla‘suc ∅)
3		peano1 7889	. . 3 ⊢ ∅ ∈ ω
4		fmlasuc 36120	. . 3 ⊢ (∅ ∈ ω → (Fmla‘suc ∅) = ((Fmla‘∅) ∪ {𝑥 ∣ ∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)}))
5	3, 4	ax-mp 5	. 2 ⊢ (Fmla‘suc ∅) = ((Fmla‘∅) ∪ {𝑥 ∣ ∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)})
6		fmla0xp 36117	. . 3 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
7		fmla0 36116	. . . . . 6 ⊢ (Fmla‘∅) = {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)}
8	7	rexeqi 3319	. . . . 5 ⊢ (∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝))
9		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑦 = 𝑝 → (𝑦 = (𝑖∈𝑔𝑗) ↔ 𝑝 = (𝑖∈𝑔𝑗)))
10	9	2rexbidv 3228	. . . . . . . . 9 ⊢ (𝑦 = 𝑝 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖∈𝑔𝑗)))
11	10	elrab 3645	. . . . . . . 8 ⊢ (𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} ↔ (𝑝 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖∈𝑔𝑗)))
12		oveq1 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → (𝑝⊼𝑔𝑞) = ((𝑖∈𝑔𝑗)⊼𝑔𝑞))
13	12	eqeq2d 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → (𝑥 = (𝑝⊼𝑔𝑞) ↔ 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞)))
14	13	rexbidv 3187	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ↔ ∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞)))
15		eqidd 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → 𝑘 = 𝑘)
16		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → 𝑝 = (𝑖∈𝑔𝑗))
17	15, 16	goaleq12d 36085	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → ∀𝑔𝑘𝑝 = ∀𝑔𝑘(𝑖∈𝑔𝑗))
18	17	eqeq2d 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → (𝑥 = ∀𝑔𝑘𝑝 ↔ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
19	18	rexbidv 3187	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → (∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝 ↔ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
20	14, 19	orbi12d 932	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
21		eqeq1 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑞 → (𝑧 = (𝑘∈𝑔𝑙) ↔ 𝑞 = (𝑘∈𝑔𝑙)))
22	21	2rexbidv 3228	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑞 → (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑧 = (𝑘∈𝑔𝑙) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑞 = (𝑘∈𝑔𝑙)))
23		fmla0 36116	. . . . . . . . . . . . . . . . . 18 ⊢ (Fmla‘∅) = {𝑧 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑧 = (𝑘∈𝑔𝑙)}
24	22, 23	elrab2 3649	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ (Fmla‘∅) ↔ (𝑞 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑞 = (𝑘∈𝑔𝑙)))
25		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑞 = (𝑘∈𝑔𝑙) → ((𝑖∈𝑔𝑗)⊼𝑔𝑞) = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)))
26	25	eqeq2d 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = (𝑘∈𝑔𝑙) → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) ↔ 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙))))
27	26	biimpcd 252	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) → (𝑞 = (𝑘∈𝑔𝑙) → 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙))))
28	27	reximdv 3178	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) → (∃𝑙 ∈ ω 𝑞 = (𝑘∈𝑔𝑙) → ∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙))))
29	28	reximdv 3178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) → (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑞 = (𝑘∈𝑔𝑙) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙))))
30	29	com12 33	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑞 = (𝑘∈𝑔𝑙) → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙))))
31	24, 30	simplbiim 514	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ (Fmla‘∅) → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙))))
32	31	rexlimiv 3157	. . . . . . . . . . . . . . 15 ⊢ (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)))
33	32	orim1i 923	. . . . . . . . . . . . . 14 ⊢ ((∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) → (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
34		r19.43 3131	. . . . . . . . . . . . . 14 ⊢ (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) ↔ (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
35	33, 34	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) → ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
36	20, 35	biimtrdi 256	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑖∈𝑔𝑗) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
37	36	com12 33	. . . . . . . . . . 11 ⊢ ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → (𝑝 = (𝑖∈𝑔𝑗) → ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
38	37	reximdv 3178	. . . . . . . . . 10 ⊢ ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → (∃𝑗 ∈ ω 𝑝 = (𝑖∈𝑔𝑗) → ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
39	38	reximdv 3178	. . . . . . . . 9 ⊢ ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖∈𝑔𝑗) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
40	39	com12 33	. . . . . . . 8 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖∈𝑔𝑗) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
41	11, 40	simplbiim 514	. . . . . . 7 ⊢ (𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))))
42	41	rexlimiv 3157	. . . . . 6 ⊢ (∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
43		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑗))
44	43	oveq1d 7427	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)))
45	44	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ↔ 𝑥 = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙))))
46	45	rexbidv 3187	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ↔ ∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙))))
47		eqidd 2762	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → 𝑘 = 𝑘)
48	47, 43	goaleq12d 36085	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → ∀𝑔𝑘(𝑖∈𝑔𝑗) = ∀𝑔𝑘(𝑚∈𝑔𝑗))
49	48	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → (𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗) ↔ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑗)))
50	46, 49	orbi12d 932	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → ((∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑗))))
51	50	rexbidv 3187	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) ↔ ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑗))))
52		oveq2 7420	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑛 → (𝑚∈𝑔𝑗) = (𝑚∈𝑔𝑛))
53	52	oveq1d 7427	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑛 → ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)))
54	53	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (𝑥 = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ↔ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙))))
55	54	rexbidv 3187	. . . . . . . . . 10 ⊢ (𝑗 = 𝑛 → (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ↔ ∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙))))
56		eqidd 2762	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑛 → 𝑘 = 𝑘)
57	56, 52	goaleq12d 36085	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → ∀𝑔𝑘(𝑚∈𝑔𝑗) = ∀𝑔𝑘(𝑚∈𝑔𝑛))
58	57	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑗 = 𝑛 → (𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑗) ↔ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)))
59	55, 58	orbi12d 932	. . . . . . . . 9 ⊢ (𝑗 = 𝑛 → ((∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑗)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛))))
60	59	rexbidv 3187	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑗)) ↔ ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛))))
61	51, 60	cbvrex2vw 3246	. . . . . . 7 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) ↔ ∃𝑚 ∈ ω ∃𝑛 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)))
62		oveq1 7419	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑜 → (𝑘∈𝑔𝑙) = (𝑜∈𝑔𝑙))
63	62	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑜 → ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)))
64	63	eqeq2d 2772	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑜 → (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ↔ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙))))
65	64	rexbidv 3187	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑜 → (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ↔ ∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙))))
66		id 23	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑜 → 𝑘 = 𝑜)
67		eqidd 2762	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑜 → (𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑛))
68	66, 67	goaleq12d 36085	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑜 → ∀𝑔𝑘(𝑚∈𝑔𝑛) = ∀𝑔𝑜(𝑚∈𝑔𝑛))
69	68	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑜 → (𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛) ↔ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)))
70	65, 69	orbi12d 932	. . . . . . . . . 10 ⊢ (𝑘 = 𝑜 → ((∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))))
71	70	cbvrexvw 3242	. . . . . . . . 9 ⊢ (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)) ↔ ∃𝑜 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)))
72	3	ne0ii 4290	. . . . . . . . . . . 12 ⊢ ω ≠ ∅
73		r19.44zv 4465	. . . . . . . . . . . 12 ⊢ (ω ≠ ∅ → (∃𝑙 ∈ ω (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))))
74	72, 73	ax-mp 5	. . . . . . . . . . 11 ⊢ (∃𝑙 ∈ ω (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)))
75		eqeq1 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑚∈𝑔𝑛) → (𝑦 = (𝑖∈𝑔𝑗) ↔ (𝑚∈𝑔𝑛) = (𝑖∈𝑔𝑗)))
76	75	2rexbidv 3228	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑚∈𝑔𝑛) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑚∈𝑔𝑛) = (𝑖∈𝑔𝑗)))
77		ovexd 7447	. . . . . . . . . . . . . . 15 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))) → (𝑚∈𝑔𝑛) ∈ V)
78		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → 𝑚 ∈ ω)
79	43	eqeq2d 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑚 → ((𝑚∈𝑔𝑛) = (𝑖∈𝑔𝑗) ↔ (𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑗)))
80	79	rexbidv 3187	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → (∃𝑗 ∈ ω (𝑚∈𝑔𝑛) = (𝑖∈𝑔𝑗) ↔ ∃𝑗 ∈ ω (𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑗)))
81	80	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑖 = 𝑚) → (∃𝑗 ∈ ω (𝑚∈𝑔𝑛) = (𝑖∈𝑔𝑗) ↔ ∃𝑗 ∈ ω (𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑗)))
82		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
83	52	eqeq2d 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑛 → ((𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑗) ↔ (𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑛)))
84	83	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑗 = 𝑛) → ((𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑗) ↔ (𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑛)))
85		eqidd 2762	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑛))
86	82, 84, 85	rspcedvd 3579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → ∃𝑗 ∈ ω (𝑚∈𝑔𝑛) = (𝑚∈𝑔𝑗))
87	78, 81, 86	rspcedvd 3579	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑚∈𝑔𝑛) = (𝑖∈𝑔𝑗))
88	87	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑚∈𝑔𝑛) = (𝑖∈𝑔𝑗))
89	76, 77, 88	elrabd 3647	. . . . . . . . . . . . . 14 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))) → (𝑚∈𝑔𝑛) ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)})
90		oveq1 7419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → (𝑝⊼𝑔𝑞) = ((𝑚∈𝑔𝑛)⊼𝑔𝑞))
91	90	eqeq2d 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → (𝑥 = (𝑝⊼𝑔𝑞) ↔ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞)))
92	91	rexbidv 3187	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ↔ ∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞)))
93		eqidd 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → 𝑘 = 𝑘)
94		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → 𝑝 = (𝑚∈𝑔𝑛))
95	93, 94	goaleq12d 36085	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → ∀𝑔𝑘𝑝 = ∀𝑔𝑘(𝑚∈𝑔𝑛))
96	95	eqeq2d 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → (𝑥 = ∀𝑔𝑘𝑝 ↔ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)))
97	96	rexbidv 3187	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → (∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝 ↔ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)))
98	92, 97	orbi12d 932	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑚∈𝑔𝑛) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛))))
99	98	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))) ∧ 𝑝 = (𝑚∈𝑔𝑛)) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛))))
100		ovexd 7447	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑜∈𝑔𝑙) ∈ V)
101		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) → 𝑜 ∈ ω)
102	101	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → 𝑜 ∈ ω)
103		oveq1 7419	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑜 → (𝑖∈𝑔𝑗) = (𝑜∈𝑔𝑗))
104	103	eqeq2d 2772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑜 → ((𝑜∈𝑔𝑙) = (𝑖∈𝑔𝑗) ↔ (𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑗)))
105	104	rexbidv 3187	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑜 → (∃𝑗 ∈ ω (𝑜∈𝑔𝑙) = (𝑖∈𝑔𝑗) ↔ ∃𝑗 ∈ ω (𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑗)))
106	105	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑖 = 𝑜) → (∃𝑗 ∈ ω (𝑜∈𝑔𝑙) = (𝑖∈𝑔𝑗) ↔ ∃𝑗 ∈ ω (𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑗)))
107		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → 𝑙 ∈ ω)
108		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑙 → (𝑜∈𝑔𝑗) = (𝑜∈𝑔𝑙))
109	108	eqeq2d 2772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → ((𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑗) ↔ (𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑙)))
110	109	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑗 = 𝑙) → ((𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑗) ↔ (𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑙)))
111		eqidd 2762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑙))
112	107, 110, 111	rspcedvd 3579	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → ∃𝑗 ∈ ω (𝑜∈𝑔𝑙) = (𝑜∈𝑔𝑗))
113	102, 106, 112	rspcedvd 3579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑜∈𝑔𝑙) = (𝑖∈𝑔𝑗))
114		eqeq1 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑜∈𝑔𝑙) → (𝑝 = (𝑖∈𝑔𝑗) ↔ (𝑜∈𝑔𝑙) = (𝑖∈𝑔𝑗)))
115	114	2rexbidv 3228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑜∈𝑔𝑙) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑜∈𝑔𝑙) = (𝑖∈𝑔𝑗)))
116		fmla0 36116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fmla‘∅) = {𝑝 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖∈𝑔𝑗)}
117	115, 116	elrab2 3649	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑜∈𝑔𝑙) ∈ (Fmla‘∅) ↔ ((𝑜∈𝑔𝑙) ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑜∈𝑔𝑙) = (𝑖∈𝑔𝑗)))
118	100, 113, 117	sylanbrc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑜∈𝑔𝑙) ∈ (Fmla‘∅))
119	118	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙))) → (𝑜∈𝑔𝑙) ∈ (Fmla‘∅))
120		oveq2 7420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = (𝑜∈𝑔𝑙) → ((𝑚∈𝑔𝑛)⊼𝑔𝑞) = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)))
121	120	eqeq2d 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = (𝑜∈𝑔𝑙) → (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞) ↔ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙))))
122	121	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙))) ∧ 𝑞 = (𝑜∈𝑔𝑙)) → (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞) ↔ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙))))
123		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙))) → 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)))
124	119, 122, 123	rspcedvd 3579	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙))) → ∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞))
125	124	ex 418	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) → ∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞)))
126	102	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) → 𝑜 ∈ ω)
127	69	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) ∧ 𝑘 = 𝑜) → (𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛) ↔ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)))
128		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) → 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))
129	126, 127, 128	rspcedvd 3579	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) → ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛))
130	129	ex 418	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛) → ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)))
131	125, 130	orim12d 979	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → ((𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) → (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛))))
132	131	imp 412	. . . . . . . . . . . . . 14 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))) → (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)))
133	89, 99, 132	rspcedvd 3579	. . . . . . . . . . . . 13 ⊢ (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛))) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝))
134	133	ex 418	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → ((𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
135	134	rexlimdva 3164	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) → (∃𝑙 ∈ ω (𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
136	74, 135	biimtrrid 246	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) → ((∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
137	136	rexlimdva 3164	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (∃𝑜 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑜∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚∈𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
138	71, 137	biimtrid 245	. . . . . . . 8 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
139	138	rexlimivv 3205	. . . . . . 7 ⊢ (∃𝑚 ∈ ω ∃𝑛 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚∈𝑔𝑛)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚∈𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝))
140	61, 139	sylbi 220	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝))
141	42, 140	impbii 212	. . . . 5 ⊢ (∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖∈𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
142	8, 141	bitri 278	. . . 4 ⊢ (∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗)))
143	142	abbii 2828	. . 3 ⊢ {𝑥 ∣ ∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))}
144	6, 143	uneq12i 4113	. 2 ⊢ ((Fmla‘∅) ∪ {𝑥 ∣ ∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)}) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))})
145	2, 5, 144	3eqtri 2788	1 ⊢ (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {cab 2739   ≠ wne 2956  ∃wrex 3087  {crab 3413  Vcvv 3451   ∪ cun 3897  ∅c0 4279  {csn 4584   × cxp 5649  suc csuc 6357  ‘cfv 6531  (class class class)co 7412  ωcom 7866  1_oc1o 8453  ∈_𝑔cgoe 36067  ⊼_𝑔cgna 36068  ∀_𝑔cgol 36069  Fmlacfmla 36071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-map 8833  df-goel 36074  df-goal 36076  df-sat 36077  df-fmla 36079

This theorem is used by:  2goelgoanfmla1  36158