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Theorem axtgpasch 28152
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 28138 . . . . . . 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 4228 . . . . . . . 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 4229 . . . . . . . 8 (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
64, 5sstri 3991 . . . . . . 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 4016 . . . . . 6 TarskiG ⊆ TarskiGB
8 axtrkg.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
97, 8sselid 3980 . . . . 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 28140 . . . . . . 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 496 . . . . . 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 1142 . . . . 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 7419 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2120eleq2d 2818 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑢 ∈ (𝑥𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑧)))
2221anbi1d 629 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧))))
23 oveq2 7420 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑣𝐼𝑥) = (𝑣𝐼𝑋))
2423eleq2d 2818 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑎 ∈ (𝑣𝐼𝑥) ↔ 𝑎 ∈ (𝑣𝐼𝑋)))
2524anbi2d 628 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)) ↔ (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
2625rexbidv 3177 . . . . . . . 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 3217 . . . . . 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 7419 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑦𝐼𝑧) = (𝑌𝐼𝑧))
3029eleq2d 2818 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑣 ∈ (𝑦𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑧)))
3130anbi2d 628 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑢 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑌𝐼𝑧))))
32 oveq2 7420 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑢𝐼𝑦) = (𝑢𝐼𝑌))
3332eleq2d 2818 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑎 ∈ (𝑢𝐼𝑦) ↔ 𝑎 ∈ (𝑢𝐼𝑌)))
3433anbi1d 629 . . . . . . . . 9 (𝑦 = 𝑌 → ((𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
3534rexbidv 3177 . . . . . . . 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 3217 . . . . . 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 7420 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
3938eleq2d 2818 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑢 ∈ (𝑋𝐼𝑧) ↔ 𝑢 ∈ (𝑋𝐼𝑍)))
40 oveq2 7420 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑌𝐼𝑧) = (𝑌𝐼𝑍))
4140eleq2d 2818 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑣 ∈ (𝑌𝐼𝑧) ↔ 𝑣 ∈ (𝑌𝐼𝑍)))
4239, 41anbi12d 630 . . . . . . . 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 3217 . . . . . 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 3627 . . . . 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 1370 . . . 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 2820 . . . . . . 7 (𝑢 = 𝑈 → (𝑢 ∈ (𝑋𝐼𝑍) ↔ 𝑈 ∈ (𝑋𝐼𝑍)))
5150anbi1d 629 . . . . . 6 (𝑢 = 𝑈 → ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍))))
52 oveq1 7419 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑢𝐼𝑌) = (𝑈𝐼𝑌))
5352eleq2d 2818 . . . . . . . 8 (𝑢 = 𝑈 → (𝑎 ∈ (𝑢𝐼𝑌) ↔ 𝑎 ∈ (𝑈𝐼𝑌)))
5453anbi1d 629 . . . . . . 7 (𝑢 = 𝑈 → ((𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))))
5554rexbidv 3177 . . . . . 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 2820 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 ∈ (𝑌𝐼𝑍) ↔ 𝑉 ∈ (𝑌𝐼𝑍)))
5857anbi2d 628 . . . . . 6 (𝑣 = 𝑉 → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) ↔ (𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍))))
59 oveq1 7419 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝐼𝑋) = (𝑉𝐼𝑋))
6059eleq2d 2818 . . . . . . . 8 (𝑣 = 𝑉 → (𝑎 ∈ (𝑣𝐼𝑋) ↔ 𝑎 ∈ (𝑉𝐼𝑋)))
6160anbi2d 628 . . . . . . 7 (𝑣 = 𝑉 → ((𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋)) ↔ (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))))
6261rexbidv 3177 . . . . . 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 3622 . . . 4 ((𝑈𝑃𝑉𝑃) → (∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑌) ∧ 𝑎 ∈ (𝑣𝐼𝑋))) → ((𝑈 ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (𝑌𝐼𝑍)) → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))))
6548, 49, 64syl2anc 583 . . 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 696 1 (𝜑 → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1085  w3a 1086   = wceq 1540  wcel 2105  {cab 2708  wral 3060  wrex 3069  {crab 3431  Vcvv 3473  [wsbc 3777  cdif 3945  cin 3947  𝒫 cpw 4602  {csn 4628  cfv 6543  (class class class)co 7412  cmpo 7414  Basecbs 17151  distcds 17213  TarskiGcstrkg 28112  TarskiGCcstrkgc 28113  TarskiGBcstrkgb 28114  TarskiGCBcstrkgcb 28115  Itvcitv 28118  LineGclng 28119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-trkgb 28134  df-trkg 28138
This theorem is referenced by:  tgbtwncom  28173  tgbtwnswapid  28177  tgbtwnintr  28178  tgtrisegint  28184  tgbtwnconn1  28260  midexlem  28377  opphllem  28420  opphllem1  28432  outpasch  28440  hlpasch  28441  lnopp2hpgb  28448  f1otrg  28556
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