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Theorem axtgpasch 26252
 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 26238 . . . . . . 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 4204 . . . . . . . 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 4205 . . . . . . . 8 (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
64, 5sstri 3975 . . . . . . 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 4000 . . . . . 6 TarskiG ⊆ TarskiGB
8 axtrkg.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
97, 8sseldi 3964 . . . . 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 26240 . . . . . . 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 499 . . . . . 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 1139 . . . . 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 7162 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2120eleq2d 2898 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑢 ∈ (𝑥𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑧)))
2221anbi1d 631 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧))))
23 oveq2 7163 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑣𝐼𝑥) = (𝑣𝐼𝑋))
2423eleq2d 2898 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑎 ∈ (𝑣𝐼𝑥) ↔ 𝑎 ∈ (𝑣𝐼𝑋)))
2524anbi2d 630 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
2625rexbidv 3297 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
2722, 26imbi12d 347 . . . . . . 7 (𝑥 = 𝑋 → (((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ↔ ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
28272ralbidv 3199 . . . . . 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 7162 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑦𝐼𝑧) = (𝑌𝐼𝑧))
3029eleq2d 2898 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑣 ∈ (𝑦𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑧)))
3130anbi2d 630 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧))))
32 oveq2 7163 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑢𝐼𝑦) = (𝑢𝐼𝑌))
3332eleq2d 2898 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑎 ∈ (𝑢𝐼𝑦) ↔ 𝑎 ∈ (𝑢𝐼𝑌)))
3433anbi1d 631 . . . . . . . . 9 (𝑦 = 𝑌 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
3534rexbidv 3297 . . . . . . . 8 (𝑦 = 𝑌 → (∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
3631, 35imbi12d 347 . . . . . . 7 (𝑦 = 𝑌 → (((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
37362ralbidv 3199 . . . . . 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 7163 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
3938eleq2d 2898 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑢 ∈ (𝑋𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑍)))
40 oveq2 7163 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑌𝐼𝑧) = (𝑌𝐼𝑍))
4140eleq2d 2898 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑣 ∈ (𝑌𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑍)))
4239, 41anbi12d 632 . . . . . . . 8 (𝑧 = 𝑍 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
4342imbi1d 344 . . . . . . 7 (𝑧 = 𝑍 → (((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
44432ralbidv 3199 . . . . . 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 3635 . . . . 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 1367 . . . 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 2900 . . . . . . 7 (𝑢 = 𝑈 → (𝑢 ∈ (𝑋𝐼𝑍) ↔ 𝑈 ∈ (𝑋𝐼𝑍)))
5150anbi1d 631 . . . . . 6 (𝑢 = 𝑈 → ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
52 oveq1 7162 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑢𝐼𝑌) = (𝑈𝐼𝑌))
5352eleq2d 2898 . . . . . . . 8 (𝑢 = 𝑈 → (𝑎 ∈ (𝑢𝐼𝑌) ↔ 𝑎 ∈ (𝑈𝐼𝑌)))
5453anbi1d 631 . . . . . . 7 (𝑢 = 𝑈 → ((𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
5554rexbidv 3297 . . . . . 6 (𝑢 = 𝑈 → (∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
5651, 55imbi12d 347 . . . . 5 (𝑢 = 𝑈 → (((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)))))
57 eleq1 2900 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 ∈ (𝑌𝐼𝑍) ↔ 𝑉 ∈ (𝑌𝐼𝑍)))
5857anbi2d 630 . . . . . 6 (𝑣 = 𝑉 → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍))))
59 oveq1 7162 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝐼𝑋) = (𝑉𝐼𝑋))
6059eleq2d 2898 . . . . . . . 8 (𝑣 = 𝑉 → (𝑎 ∈ (𝑣𝐼𝑋) ↔ 𝑎 ∈ (𝑉𝐼𝑋)))
6160anbi2d 630 . . . . . . 7 (𝑣 = 𝑉 → ((𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
6261rexbidv 3297 . . . . . 6 (𝑣 = 𝑉 → (∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
6358, 62imbi12d 347 . . . . 5 (𝑣 = 𝑉 → (((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) ↔ ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))))
6456, 63rspc2v 3632 . . . 4 ((𝑈𝑃𝑉𝑃) → (∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))))
6548, 49, 64syl2anc 586 . . 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 398   ∨ w3o 1082   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494  [wsbc 3771   ∖ cdif 3932   ∩ cin 3934  𝒫 cpw 4538  {csn 4566  ‘cfv 6354  (class class class)co 7155   ∈ cmpo 7157  Basecbs 16482  distcds 16573  TarskiGcstrkg 26215  TarskiGCcstrkgc 26216  TarskiGBcstrkgb 26217  TarskiGCBcstrkgcb 26218  Itvcitv 26221  LineGclng 26222 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158  df-trkgb 26234  df-trkg 26238 This theorem is referenced by:  tgbtwncom  26273  tgbtwnswapid  26277  tgbtwnintr  26278  tgtrisegint  26284  tgbtwnconn1  26360  midexlem  26477  opphllem  26520  opphllem1  26532  outpasch  26540  hlpasch  26541  lnopp2hpgb  26548  f1otrg  26656
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