Theorem axtgpasch 26828

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 26814	. . . . . . 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 4162	. . . . . . . 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 4163	. . . . . . . 8 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
6	4, 5	sstri 3930	. . . . . . 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 3955	. . . . . 6 ⊢ TarskiG ⊆ TarskiGB
8		axtrkg.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	7, 8	sselid 3919	. . . . 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 26816	. . . . . . 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 497	. . . . . 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 1142	. . . . 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 7282	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
21	20	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑢 ∈ (𝑥𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑧)))
22	21	anbi1d 630	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧))))
23		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑣𝐼𝑥) = (𝑣𝐼𝑋))
24	23	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑎 ∈ (𝑣𝐼𝑥) ↔ 𝑎 ∈ (𝑣𝐼𝑋)))
25	24	anbi2d 629	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
26	25	rexbidv 3226	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
27	22, 26	imbi12d 345	. . . . . . 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 3129	. . . . . 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 7282	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦𝐼𝑧) = (𝑌𝐼𝑧))
30	29	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑣 ∈ (𝑦𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑧)))
31	30	anbi2d 629	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧))))
32		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝑢𝐼𝑦) = (𝑢𝐼𝑌))
33	32	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑎 ∈ (𝑢𝐼𝑦) ↔ 𝑎 ∈ (𝑢𝐼𝑌)))
34	33	anbi1d 630	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
35	34	rexbidv 3226	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
36	31, 35	imbi12d 345	. . . . . . 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 3129	. . . . . 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 7283	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
39	38	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (𝑢 ∈ (𝑋𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑍)))
40		oveq2 7283	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (𝑌𝐼𝑧) = (𝑌𝐼𝑍))
41	40	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (𝑣 ∈ (𝑌𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑍)))
42	39, 41	anbi12d 631	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
43	42	imbi1d 342	. . . . . . 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 3129	. . . . . 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 3573	. . . . 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 1370	. . . 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 2826	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (𝑋𝐼𝑍) ↔ 𝑈 ∈ (𝑋𝐼𝑍)))
51	50	anbi1d 630	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
52		oveq1 7282	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑢𝐼𝑌) = (𝑈𝐼𝑌))
53	52	eleq2d 2824	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑎 ∈ (𝑢𝐼𝑌) ↔ 𝑎 ∈ (𝑈𝐼𝑌)))
54	53	anbi1d 630	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
55	54	rexbidv 3226	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
56	51, 55	imbi12d 345	. . . . 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 2826	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (𝑣 ∈ (𝑌𝐼𝑍) ↔ 𝑉 ∈ (𝑌𝐼𝑍)))
58	57	anbi2d 629	. . . . . 6 ⊢ (𝑣 = 𝑉 → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍))))
59		oveq1 7282	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑣𝐼𝑋) = (𝑉𝐼𝑋))
60	59	eleq2d 2824	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑎 ∈ (𝑣𝐼𝑋) ↔ 𝑎 ∈ (𝑉𝐼𝑋)))
61	60	anbi2d 629	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
62	61	rexbidv 3226	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
63	58, 62	imbi12d 345	. . . . 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 3570	. . . 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 584	. . 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 696	1 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))

Syntax hints:   → wi 4   ∧ wa 396   ∨ w3o 1085   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432  [wsbc 3716   ∖ cdif 3884   ∩ cin 3886  𝒫 cpw 4533  {csn 4561  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  TarskiG_Ccstrkgc 26789  TarskiG_Bcstrkgb 26790  TarskiG_CBcstrkgcb 26791  Itvcitv 26794  LineGclng 26795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-trkgb 26810  df-trkg 26814

This theorem is referenced by:  tgbtwncom  26849  tgbtwnswapid  26853  tgbtwnintr  26854  tgtrisegint  26860  tgbtwnconn1  26936  midexlem  27053  opphllem  27096  opphllem1  27108  outpasch  27116  hlpasch  27117  lnopp2hpgb  27124  f1otrg  27232