Theorem axtgpasch 27985

Description: Axiom of (Inner) Pasch, Axiom A7 of [Schwabhauser] p. 12. Given triangle 𝑋𝑌𝑍, point 𝑈 in segment 𝑋𝑍, and point 𝑉 in segment 𝑌𝑍, there exists a point 𝑎 on both the segment 𝑈𝑌 and the segment 𝑉𝑋. This axiom is essentially a subset of the general Pasch axiom. The general Pasch axiom asserts that on a plane "a line intersecting a triangle in one of its sides, and not intersecting any of the vertices, must intersect one of the other two sides" (per the discussion about Axiom 7 of [Tarski1999] p. 179). The (general) Pasch axiom was used implicitly by Euclid, but never stated; Moritz Pasch discovered its omission in 1882. As noted in the Metamath book, this means that the omission of Pasch's axiom from Euclid went unnoticed for 2000 years. Only the inner Pasch algorithm is included as an axiom; the "outer" form of the Pasch axiom can be proved using the inner form (see theorem 9.6 of [Schwabhauser] p. 69 and the brief discussion in axiom 7.1 of [Tarski1999] p. 180). (Contributed by Thierry Arnoux, 15-Mar-2019.)

Hypotheses

Ref	Expression
axtrkg.p	⊢ 𝑃 = (Base‘𝐺)
axtrkg.d	⊢ − = (dist‘𝐺)
axtrkg.i	⊢ 𝐼 = (Itv‘𝐺)
axtrkg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
axtgpasch.1	⊢ (𝜑 → 𝑋 ∈ 𝑃)
axtgpasch.2	⊢ (𝜑 → 𝑌 ∈ 𝑃)
axtgpasch.3	⊢ (𝜑 → 𝑍 ∈ 𝑃)
axtgpasch.4	⊢ (𝜑 → 𝑈 ∈ 𝑃)
axtgpasch.5	⊢ (𝜑 → 𝑉 ∈ 𝑃)
axtgpasch.6	⊢ (𝜑 → 𝑈 ∈ (𝑋𝐼𝑍))
axtgpasch.7	⊢ (𝜑 → 𝑉 ∈ (𝑌𝐼𝑍))

Assertion

Ref	Expression
axtgpasch	⊢ (𝜑 → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))

Distinct variable groups:   𝐼,𝑎   𝑃,𝑎   𝑈,𝑎   𝑋,𝑎   𝑌,𝑎   𝑍,𝑎   𝑉,𝑎   − ,𝑎

Allowed substitution hints:   𝜑(𝑎)   𝐺(𝑎)

Proof of Theorem axtgpasch

Dummy variables 𝑓 𝑖 𝑝 𝑥 𝑦 𝑧 𝑏 𝑣 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axtgpasch.6	. 2 ⊢ (𝜑 → 𝑈 ∈ (𝑋𝐼𝑍))
2		axtgpasch.7	. 2 ⊢ (𝜑 → 𝑉 ∈ (𝑌𝐼𝑍))
3		df-trkg 27971	. . . . . . 7 ⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
4		inss1 4227	. . . . . . . 8 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGC ∩ TarskiGB)
5		inss2 4228	. . . . . . . 8 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
6	4, 5	sstri 3990	. . . . . . 7 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGB
7	3, 6	eqsstri 4015	. . . . . 6 ⊢ TarskiG ⊆ TarskiGB
8		axtrkg.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	7, 8	sselid 3979	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiGB)
10		axtrkg.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
11		axtrkg.d	. . . . . . . 8 ⊢ − = (dist‘𝐺)
12		axtrkg.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
13	10, 11, 12	istrkgb 27973	. . . . . . 7 ⊢ (𝐺 ∈ TarskiGB ↔ (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)))))
14	13	simprbi 495	. . . . . 6 ⊢ (𝐺 ∈ TarskiGB → (∀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))))
15	14	simp2d 1141	. . . . 5 ⊢ (𝐺 ∈ TarskiGB → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))))
16	9, 15	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))))
17		axtgpasch.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
18		axtgpasch.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
19		axtgpasch.3	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
20		oveq1 7418	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
21	20	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑢 ∈ (𝑥𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑧)))
22	21	anbi1d 628	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧))))
23		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑣𝐼𝑥) = (𝑣𝐼𝑋))
24	23	eleq2d 2817	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑎 ∈ (𝑣𝐼𝑥) ↔ 𝑎 ∈ (𝑣𝐼𝑋)))
25	24	anbi2d 627	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
26	25	rexbidv 3176	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
27	22, 26	imbi12d 343	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
28	27	2ralbidv 3216	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∀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)))))
29		oveq1 7418	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦𝐼𝑧) = (𝑌𝐼𝑧))
30	29	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑣 ∈ (𝑦𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑧)))
31	30	anbi2d 627	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧))))
32		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝑢𝐼𝑦) = (𝑢𝐼𝑌))
33	32	eleq2d 2817	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑎 ∈ (𝑢𝐼𝑦) ↔ 𝑎 ∈ (𝑢𝐼𝑌)))
34	33	anbi1d 628	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
35	34	rexbidv 3176	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
36	31, 35	imbi12d 343	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
37	36	2ralbidv 3216	. . . . . 6 ⊢ (𝑦 = 𝑌 → (∀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)))))
38		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
39	38	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (𝑢 ∈ (𝑋𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑍)))
40		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (𝑌𝐼𝑧) = (𝑌𝐼𝑍))
41	40	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (𝑣 ∈ (𝑌𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑍)))
42	39, 41	anbi12d 629	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
43	42	imbi1d 340	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
44	43	2ralbidv 3216	. . . . . 6 ⊢ (𝑧 = 𝑍 → (∀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)))))
45	28, 37, 44	rspc3v 3626	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → (∀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)))))
46	17, 18, 19, 45	syl3anc 1369	. . . 4 ⊢ (𝜑 → (∀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)))))
47	16, 46	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
48		axtgpasch.4	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
49		axtgpasch.5	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
50		eleq1 2819	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (𝑋𝐼𝑍) ↔ 𝑈 ∈ (𝑋𝐼𝑍)))
51	50	anbi1d 628	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
52		oveq1 7418	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑢𝐼𝑌) = (𝑈𝐼𝑌))
53	52	eleq2d 2817	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑎 ∈ (𝑢𝐼𝑌) ↔ 𝑎 ∈ (𝑈𝐼𝑌)))
54	53	anbi1d 628	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
55	54	rexbidv 3176	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
56	51, 55	imbi12d 343	. . . . 5 ⊢ (𝑢 = 𝑈 → (((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
57		eleq1 2819	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (𝑣 ∈ (𝑌𝐼𝑍) ↔ 𝑉 ∈ (𝑌𝐼𝑍)))
58	57	anbi2d 627	. . . . . 6 ⊢ (𝑣 = 𝑉 → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍))))
59		oveq1 7418	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑣𝐼𝑋) = (𝑉𝐼𝑋))
60	59	eleq2d 2817	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑎 ∈ (𝑣𝐼𝑋) ↔ 𝑎 ∈ (𝑉𝐼𝑋)))
61	60	anbi2d 627	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
62	61	rexbidv 3176	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
63	58, 62	imbi12d 343	. . . . 5 ⊢ (𝑣 = 𝑉 → (((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))))
64	56, 63	rspc2v 3621	. . . 4 ⊢ ((𝑈 ∈ 𝑃 ∧ 𝑉 ∈ 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)))))
65	48, 49, 64	syl2anc 582	. . 3 ⊢ (𝜑 → (∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))))
66	47, 65	mpd 15	. 2 ⊢ (𝜑 → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
67	1, 2, 66	mp2and 695	1 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))

Syntax hints:   → wi 4   ∧ wa 394   ∨ w3o 1084   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  {cab 2707  ∀wral 3059  ∃wrex 3068  {crab 3430  Vcvv 3472  [wsbc 3776   ∖ cdif 3944   ∩ cin 3946  𝒫 cpw 4601  {csn 4627  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17148  distcds 17210  TarskiGcstrkg 27945  TarskiG_Ccstrkgc 27946  TarskiG_Bcstrkgb 27947  TarskiG_CBcstrkgcb 27948  Itvcitv 27951  LineGclng 27952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-trkgb 27967  df-trkg 27971

This theorem is referenced by:  tgbtwncom  28006  tgbtwnswapid  28010  tgbtwnintr  28011  tgtrisegint  28017  tgbtwnconn1  28093  midexlem  28210  opphllem  28253  opphllem1  28265  outpasch  28273  hlpasch  28274  lnopp2hpgb  28281  f1otrg  28389