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Theorem istrkgb 25154

Description: Property of being a Tarski geometry - betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)

**Hypotheses**
Ref	Expression
istrkg.p	⊢ 𝑃 = (Base‘𝐺)
istrkg.d	⊢ − = (dist‘𝐺)
istrkg.i	⊢ 𝐼 = (Itv‘𝐺)

**Assertion**
Ref	Expression
istrkgb	⊢ (𝐺 ∈ TarskiG_B ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))

Distinct variable groups: 𝑎,𝑏,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧,𝐼 𝑃,𝑎,𝑏,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧 − ,𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐺(𝑥,𝑦,𝑧,𝑣,𝑢,𝑡,𝑠,𝑎,𝑏) − (𝑡,𝑠)

Proof of Theorem istrkgb Dummy variables 𝑓 𝑖 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		istrkg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		istrkg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
3		simpl 472	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑝 = 𝑃)
4	3	eqcomd 2616	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑃 = 𝑝)
5	4	adantr 480	. . . . . 6 ⊢ (((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) → 𝑃 = 𝑝)
6		simpllr 795	. . . . . . . . . 10 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → 𝑖 = 𝐼)
7	6	eqcomd 2616	. . . . . . . . 9 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → 𝐼 = 𝑖)
8	7	oveqd 6566	. . . . . . . 8 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → (𝑥𝐼𝑥) = (𝑥𝑖𝑥))
9	8	eleq2d 2673	. . . . . . 7 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → (𝑦 ∈ (𝑥𝐼𝑥) ↔ 𝑦 ∈ (𝑥𝑖𝑥)))
10	9	imbi1d 330	. . . . . 6 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → ((𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ↔ (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦)))
11	5, 10	raleqbidva 3131	. . . . 5 ⊢ (((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) → (∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦)))
12	4, 11	raleqbidva 3131	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦)))
13	5	adantr 480	. . . . . . 7 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → 𝑃 = 𝑝)
14	13	adantr 480	. . . . . . . 8 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → 𝑃 = 𝑝)
15	14	adantr 480	. . . . . . . . 9 ⊢ ((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) → 𝑃 = 𝑝)
16		simp-6r 807	. . . . . . . . . . . . . 14 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → 𝑖 = 𝐼)
17	16	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → 𝐼 = 𝑖)
18	17	oveqd 6566	. . . . . . . . . . . 12 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → (𝑥𝐼𝑧) = (𝑥𝑖𝑧))
19	18	eleq2d 2673	. . . . . . . . . . 11 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → (𝑢 ∈ (𝑥𝐼𝑧) ↔ 𝑢 ∈ (𝑥𝑖𝑧)))
20	17	oveqd 6566	. . . . . . . . . . . 12 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → (𝑦𝐼𝑧) = (𝑦𝑖𝑧))
21	20	eleq2d 2673	. . . . . . . . . . 11 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → (𝑣 ∈ (𝑦𝐼𝑧) ↔ 𝑣 ∈ (𝑦𝑖𝑧)))
22	19, 21	anbi12d 743	. . . . . . . . . 10 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧))))
23	15	adantr 480	. . . . . . . . . . 11 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → 𝑃 = 𝑝)
24	17	oveqdr 6573	. . . . . . . . . . . . 13 ⊢ ((((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑎 ∈ 𝑃) → (𝑢𝐼𝑦) = (𝑢𝑖𝑦))
25	24	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑎 ∈ 𝑃) → (𝑎 ∈ (𝑢𝐼𝑦) ↔ 𝑎 ∈ (𝑢𝑖𝑦)))
26	17	oveqdr 6573	. . . . . . . . . . . . 13 ⊢ ((((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑎 ∈ 𝑃) → (𝑣𝐼𝑥) = (𝑣𝑖𝑥))
27	26	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑎 ∈ 𝑃) → (𝑎 ∈ (𝑣𝐼𝑥) ↔ 𝑎 ∈ (𝑣𝑖𝑥)))
28	25, 27	anbi12d 743	. . . . . . . . . . 11 ⊢ ((((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑎 ∈ 𝑃) → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))))
29	23, 28	rexeqbidva 3132	. . . . . . . . . 10 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))))
30	22, 29	imbi12d 333	. . . . . . . . 9 ⊢ (((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) → (((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥)))))
31	15, 30	raleqbidva 3131	. . . . . . . 8 ⊢ ((((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ 𝑢 ∈ 𝑃) → (∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥)))))
32	14, 31	raleqbidva 3131	. . . . . . 7 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥)))))
33	13, 32	raleqbidva 3131	. . . . . 6 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → (∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥)))))
34	5, 33	raleqbidva 3131	. . . . 5 ⊢ (((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑥 ∈ 𝑃) → (∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥)))))
35	4, 34	raleqbidva 3131	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥)))))
36	4	pweqd 4113	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝒫 𝑃 = 𝒫 𝑝)
37	36	adantr 480	. . . . . 6 ⊢ (((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) → 𝒫 𝑃 = 𝒫 𝑝)
38	4	ad2antrr 758	. . . . . . . 8 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) → 𝑃 = 𝑝)
39		simp-4r 803	. . . . . . . . . . . 12 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑎 ∈ 𝑃) → 𝑖 = 𝐼)
40	39	eqcomd 2616	. . . . . . . . . . 11 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑎 ∈ 𝑃) → 𝐼 = 𝑖)
41	40	oveqd 6566	. . . . . . . . . 10 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑎 ∈ 𝑃) → (𝑎𝐼𝑦) = (𝑎𝑖𝑦))
42	41	eleq2d 2673	. . . . . . . . 9 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑎 ∈ 𝑃) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑥 ∈ (𝑎𝑖𝑦)))
43	42	2ralbidv 2972	. . . . . . . 8 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑎 ∈ 𝑃) → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦)))
44	38, 43	rexeqbidva 3132	. . . . . . 7 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦)))
45		simp-4r 803	. . . . . . . . . . . 12 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑏 ∈ 𝑃) → 𝑖 = 𝐼)
46	45	eqcomd 2616	. . . . . . . . . . 11 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑏 ∈ 𝑃) → 𝐼 = 𝑖)
47	46	oveqd 6566	. . . . . . . . . 10 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑏 ∈ 𝑃) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
48	47	eleq2d 2673	. . . . . . . . 9 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑏 ∈ 𝑃) → (𝑏 ∈ (𝑥𝐼𝑦) ↔ 𝑏 ∈ (𝑥𝑖𝑦)))
49	48	2ralbidv 2972	. . . . . . . 8 ⊢ (((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) ∧ 𝑏 ∈ 𝑃) → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))
50	38, 49	rexeqbidva 3132	. . . . . . 7 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) → (∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))
51	44, 50	imbi12d 333	. . . . . 6 ⊢ ((((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑡 ∈ 𝒫 𝑃) → ((∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ (∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦))))
52	37, 51	raleqbidva 3131	. . . . 5 ⊢ (((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) ∧ 𝑠 ∈ 𝒫 𝑃) → (∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ ∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦))))
53	36, 52	raleqbidva 3131	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦))))
54	12, 35, 53	3anbi123d 1391	. . 3 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦))) ↔ (∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))))
55	1, 2, 54	sbcie2s 15744	. 2 ⊢ (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦))) ↔ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
56		df-trkgb 25148	. 2 ⊢ TarskiG_B = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))}
57	55, 56	elab4g 3324	1 ⊢ (𝐺 ∈ TarskiG_B ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 [wsbc 3402 𝒫 cpw 4108 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiG_Bcstrkgb 25131 Itvcitv 25135

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-trkgb 25148

This theorem is referenced by: axtgbtwnid 25165 axtgpasch 25166 axtgcont1 25167 f1otrg 25551 eengtrkg 25665

W3C validator