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Theorem axtgpasch 27409
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 (𝜑𝐺 ∈ TarskiG)
axtgpasch.1 (𝜑𝑋𝑃)
axtgpasch.2 (𝜑𝑌𝑃)
axtgpasch.3 (𝜑𝑍𝑃)
axtgpasch.4 (𝜑𝑈𝑃)
axtgpasch.5 (𝜑𝑉𝑃)
axtgpasch.6 (𝜑𝑈 ∈ (𝑋𝐼𝑍))
axtgpasch.7 (𝜑𝑉 ∈ (𝑌𝐼𝑍))
Assertion
Ref Expression
axtgpasch (𝜑 → ∃𝑎𝑃 (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.
StepHypRef Expression
1 axtgpasch.6 . 2 (𝜑𝑈 ∈ (𝑋𝐼𝑍))
2 axtgpasch.7 . 2 (𝜑𝑉 ∈ (𝑌𝐼𝑍))
3 df-trkg 27395 . . . . . . 7 TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
4 inss1 4188 . . . . . . . 8 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGC ∩ TarskiGB)
5 inss2 4189 . . . . . . . 8 (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
64, 5sstri 3953 . . . . . . 7 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGB
73, 6eqsstri 3978 . . . . . 6 TarskiG ⊆ TarskiGB
8 axtrkg.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
97, 8sselid 3942 . . . . 5 (𝜑𝐺 ∈ TarskiGB)
10 axtrkg.p . . . . . . . 8 𝑃 = (Base‘𝐺)
11 axtrkg.d . . . . . . . 8 = (dist‘𝐺)
12 axtrkg.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
1310, 11, 12istrkgb 27397 . . . . . . 7 (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
1413simprbi 497 . . . . . 6 (𝐺 ∈ TarskiGB → (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
1514simp2d 1143 . . . . 5 (𝐺 ∈ TarskiGB → ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))))
169, 15syl 17 . . . 4 (𝜑 → ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))))
17 axtgpasch.1 . . . . 5 (𝜑𝑋𝑃)
18 axtgpasch.2 . . . . 5 (𝜑𝑌𝑃)
19 axtgpasch.3 . . . . 5 (𝜑𝑍𝑃)
20 oveq1 7364 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2120eleq2d 2823 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑢 ∈ (𝑥𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑧)))
2221anbi1d 630 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧))))
23 oveq2 7365 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑣𝐼𝑥) = (𝑣𝐼𝑋))
2423eleq2d 2823 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑎 ∈ (𝑣𝐼𝑥) ↔ 𝑎 ∈ (𝑣𝐼𝑋)))
2524anbi2d 629 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
2625rexbidv 3175 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
2722, 26imbi12d 344 . . . . . . 7 (𝑥 = 𝑋 → (((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
28272ralbidv 3212 . . . . . 6 (𝑥 = 𝑋 → (∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
29 oveq1 7364 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑦𝐼𝑧) = (𝑌𝐼𝑧))
3029eleq2d 2823 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑣 ∈ (𝑦𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑧)))
3130anbi2d 629 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧))))
32 oveq2 7365 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑢𝐼𝑦) = (𝑢𝐼𝑌))
3332eleq2d 2823 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑎 ∈ (𝑢𝐼𝑦) ↔ 𝑎 ∈ (𝑢𝐼𝑌)))
3433anbi1d 630 . . . . . . . . 9 (𝑦 = 𝑌 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
3534rexbidv 3175 . . . . . . . 8 (𝑦 = 𝑌 → (∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
3631, 35imbi12d 344 . . . . . . 7 (𝑦 = 𝑌 → (((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
37362ralbidv 3212 . . . . . 6 (𝑦 = 𝑌 → (∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
38 oveq2 7365 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
3938eleq2d 2823 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑢 ∈ (𝑋𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑍)))
40 oveq2 7365 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑌𝐼𝑧) = (𝑌𝐼𝑍))
4140eleq2d 2823 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑣 ∈ (𝑌𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑍)))
4239, 41anbi12d 631 . . . . . . . 8 (𝑧 = 𝑍 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
4342imbi1d 341 . . . . . . 7 (𝑧 = 𝑍 → (((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
44432ralbidv 3212 . . . . . 6 (𝑧 = 𝑍 → (∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
4528, 37, 44rspc3v 3593 . . . . 5 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) → ∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
4617, 18, 19, 45syl3anc 1371 . . . 4 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) → ∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
4716, 46mpd 15 . . 3 (𝜑 → ∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
48 axtgpasch.4 . . . 4 (𝜑𝑈𝑃)
49 axtgpasch.5 . . . 4 (𝜑𝑉𝑃)
50 eleq1 2825 . . . . . . 7 (𝑢 = 𝑈 → (𝑢 ∈ (𝑋𝐼𝑍) ↔ 𝑈 ∈ (𝑋𝐼𝑍)))
5150anbi1d 630 . . . . . 6 (𝑢 = 𝑈 → ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
52 oveq1 7364 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑢𝐼𝑌) = (𝑈𝐼𝑌))
5352eleq2d 2823 . . . . . . . 8 (𝑢 = 𝑈 → (𝑎 ∈ (𝑢𝐼𝑌) ↔ 𝑎 ∈ (𝑈𝐼𝑌)))
5453anbi1d 630 . . . . . . 7 (𝑢 = 𝑈 → ((𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
5554rexbidv 3175 . . . . . 6 (𝑢 = 𝑈 → (∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
5651, 55imbi12d 344 . . . . 5 (𝑢 = 𝑈 → (((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
57 eleq1 2825 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 ∈ (𝑌𝐼𝑍) ↔ 𝑉 ∈ (𝑌𝐼𝑍)))
5857anbi2d 629 . . . . . 6 (𝑣 = 𝑉 → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍))))
59 oveq1 7364 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝐼𝑋) = (𝑉𝐼𝑋))
6059eleq2d 2823 . . . . . . . 8 (𝑣 = 𝑉 → (𝑎 ∈ (𝑣𝐼𝑋) ↔ 𝑎 ∈ (𝑉𝐼𝑋)))
6160anbi2d 629 . . . . . . 7 (𝑣 = 𝑉 → ((𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
6261rexbidv 3175 . . . . . 6 (𝑣 = 𝑉 → (∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
6358, 62imbi12d 344 . . . . 5 (𝑣 = 𝑉 → (((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))))
6456, 63rspc2v 3590 . . . 4 ((𝑈𝑃𝑉𝑃) → (∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))))
6548, 49, 64syl2anc 584 . . 3 (𝜑 → (∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))))
6647, 65mpd 15 . 2 (𝜑 → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
671, 2, 66mp2and 697 1 (𝜑 → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  [wsbc 3739  cdif 3907  cin 3909  𝒫 cpw 4560  {csn 4586  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  TarskiGCcstrkgc 27370  TarskiGBcstrkgb 27371  TarskiGCBcstrkgcb 27372  Itvcitv 27375  LineGclng 27376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-trkgb 27391  df-trkg 27395
This theorem is referenced by:  tgbtwncom  27430  tgbtwnswapid  27434  tgbtwnintr  27435  tgtrisegint  27441  tgbtwnconn1  27517  midexlem  27634  opphllem  27677  opphllem1  27689  outpasch  27697  hlpasch  27698  lnopp2hpgb  27705  f1otrg  27813
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