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Theorem axtg5seg 28473
Description: Five segments axiom, Axiom A5 of [Schwabhauser] p. 11. Take two triangles 𝑋𝑍𝑈 and 𝐴𝐶𝑉, a point 𝑌 on 𝑋𝑍, and a point 𝐵 on 𝐴𝐶. If all corresponding line segments except for 𝑍𝑈 and 𝐶𝑉 are congruent ( i.e., 𝑋𝑌 𝐴𝐵, 𝑌𝑍 𝐵𝐶, 𝑋𝑈 𝐴𝑉, and 𝑌𝑈 𝐵𝑉), then 𝑍𝑈 and 𝐶𝑉 are also congruent. As noted in Axiom 5 of [Tarski1999] p. 178, "this axiom is similar in character to the well-known theorems of Euclidean geometry that allow one to conclude, from hypotheses about the congruence of certain corresponding sides and angles in two triangles, the congruence of other corresponding sides and angles." (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p 𝑃 = (Base‘𝐺)
axtrkg.d = (dist‘𝐺)
axtrkg.i 𝐼 = (Itv‘𝐺)
axtrkg.g (𝜑𝐺 ∈ TarskiG)
axtg5seg.1 (𝜑𝑋𝑃)
axtg5seg.2 (𝜑𝑌𝑃)
axtg5seg.3 (𝜑𝑍𝑃)
axtg5seg.4 (𝜑𝐴𝑃)
axtg5seg.5 (𝜑𝐵𝑃)
axtg5seg.6 (𝜑𝐶𝑃)
axtg5seg.7 (𝜑𝑈𝑃)
axtg5seg.8 (𝜑𝑉𝑃)
axtg5seg.9 (𝜑𝑋𝑌)
axtg5seg.10 (𝜑𝑌 ∈ (𝑋𝐼𝑍))
axtg5seg.11 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
axtg5seg.12 (𝜑 → (𝑋 𝑌) = (𝐴 𝐵))
axtg5seg.13 (𝜑 → (𝑌 𝑍) = (𝐵 𝐶))
axtg5seg.14 (𝜑 → (𝑋 𝑈) = (𝐴 𝑉))
axtg5seg.15 (𝜑 → (𝑌 𝑈) = (𝐵 𝑉))
Assertion
Ref Expression
axtg5seg (𝜑 → (𝑍 𝑈) = (𝐶 𝑉))

Proof of Theorem axtg5seg
Dummy variables 𝑓 𝑖 𝑝 𝑥 𝑦 𝑧 𝑎 𝑏 𝑐 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 28461 . . . . . . 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))})}))
2 inss2 4238 . . . . . . . 8 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
3 inss1 4237 . . . . . . . 8 (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}) ⊆ TarskiGCB
42, 3sstri 3993 . . . . . . 7 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGCB
51, 4eqsstri 4030 . . . . . 6 TarskiG ⊆ TarskiGCB
6 axtrkg.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
75, 6sselid 3981 . . . . 5 (𝜑𝐺 ∈ TarskiGCB)
8 axtrkg.p . . . . . . . 8 𝑃 = (Base‘𝐺)
9 axtrkg.d . . . . . . . 8 = (dist‘𝐺)
10 axtrkg.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
118, 9, 10istrkgcb 28464 . . . . . . 7 (𝐺 ∈ TarskiGCB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ∧ ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)))))
1211simprbi 496 . . . . . 6 (𝐺 ∈ TarskiGCB → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ∧ ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
1312simpld 494 . . . . 5 (𝐺 ∈ TarskiGCB → ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)))
147, 13syl 17 . . . 4 (𝜑 → ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)))
15 axtg5seg.1 . . . . 5 (𝜑𝑋𝑃)
16 axtg5seg.2 . . . . 5 (𝜑𝑌𝑃)
17 axtg5seg.3 . . . . 5 (𝜑𝑍𝑃)
18 neeq1 3003 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
19 oveq1 7438 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2019eleq2d 2827 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
2118, 203anbi12d 1439 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
22 oveq1 7438 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
2322eqeq1d 2739 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑎 𝑏) ↔ (𝑋 𝑦) = (𝑎 𝑏)))
2423anbi1d 631 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐))))
25 oveq1 7438 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 𝑢) = (𝑋 𝑢))
2625eqeq1d 2739 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 𝑢) = (𝑎 𝑣) ↔ (𝑋 𝑢) = (𝑎 𝑣)))
2726anbi1d 631 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))))
2824, 27anbi12d 632 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))) ↔ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))))
2921, 28anbi12d 632 . . . . . . . . . 10 (𝑥 = 𝑋 → (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) ↔ ((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))))))
3029imbi1d 341 . . . . . . . . 9 (𝑥 = 𝑋 → ((((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
3130ralbidv 3178 . . . . . . . 8 (𝑥 = 𝑋 → (∀𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
32312ralbidv 3221 . . . . . . 7 (𝑥 = 𝑋 → (∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
33322ralbidv 3221 . . . . . 6 (𝑥 = 𝑋 → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
34 neeq2 3004 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋𝑦𝑋𝑌))
35 eleq1 2829 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
3634, 353anbi12d 1439 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
37 oveq2 7439 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
3837eqeq1d 2739 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑎 𝑏) ↔ (𝑋 𝑌) = (𝑎 𝑏)))
39 oveq1 7438 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
4039eqeq1d 2739 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑦 𝑧) = (𝑏 𝑐) ↔ (𝑌 𝑧) = (𝑏 𝑐)))
4138, 40anbi12d 632 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐))))
42 oveq1 7438 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑦 𝑢) = (𝑌 𝑢))
4342eqeq1d 2739 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑦 𝑢) = (𝑏 𝑣) ↔ (𝑌 𝑢) = (𝑏 𝑣)))
4443anbi2d 630 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))))
4541, 44anbi12d 632 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))))
4636, 45anbi12d 632 . . . . . . . . . 10 (𝑦 = 𝑌 → (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))))))
4746imbi1d 341 . . . . . . . . 9 (𝑦 = 𝑌 → ((((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
4847ralbidv 3178 . . . . . . . 8 (𝑦 = 𝑌 → (∀𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
49482ralbidv 3221 . . . . . . 7 (𝑦 = 𝑌 → (∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
50492ralbidv 3221 . . . . . 6 (𝑦 = 𝑌 → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
51 oveq2 7439 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5251eleq2d 2827 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
53523anbi2d 1443 . . . . . . . . . . 11 (𝑧 = 𝑍 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
54 oveq2 7439 . . . . . . . . . . . . . 14 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
5554eqeq1d 2739 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → ((𝑌 𝑧) = (𝑏 𝑐) ↔ (𝑌 𝑍) = (𝑏 𝑐)))
5655anbi2d 630 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐))))
5756anbi1d 631 . . . . . . . . . . 11 (𝑧 = 𝑍 → ((((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))))
5853, 57anbi12d 632 . . . . . . . . . 10 (𝑧 = 𝑍 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))))))
59 oveq1 7438 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑢) = (𝑍 𝑢))
6059eqeq1d 2739 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑧 𝑢) = (𝑐 𝑣) ↔ (𝑍 𝑢) = (𝑐 𝑣)))
6158, 60imbi12d 344 . . . . . . . . 9 (𝑧 = 𝑍 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
6261ralbidv 3178 . . . . . . . 8 (𝑧 = 𝑍 → (∀𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
63622ralbidv 3221 . . . . . . 7 (𝑧 = 𝑍 → (∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
64632ralbidv 3221 . . . . . 6 (𝑧 = 𝑍 → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
6533, 50, 64rspc3v 3638 . . . . 5 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) → ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
6615, 16, 17, 65syl3anc 1373 . . . 4 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) → ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
6714, 66mpd 15 . . 3 (𝜑 → ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)))
68 axtg5seg.7 . . . 4 (𝜑𝑈𝑃)
69 axtg5seg.4 . . . 4 (𝜑𝐴𝑃)
70 axtg5seg.5 . . . 4 (𝜑𝐵𝑃)
71 oveq2 7439 . . . . . . . . . . 11 (𝑢 = 𝑈 → (𝑋 𝑢) = (𝑋 𝑈))
7271eqeq1d 2739 . . . . . . . . . 10 (𝑢 = 𝑈 → ((𝑋 𝑢) = (𝑎 𝑣) ↔ (𝑋 𝑈) = (𝑎 𝑣)))
73 oveq2 7439 . . . . . . . . . . 11 (𝑢 = 𝑈 → (𝑌 𝑢) = (𝑌 𝑈))
7473eqeq1d 2739 . . . . . . . . . 10 (𝑢 = 𝑈 → ((𝑌 𝑢) = (𝑏 𝑣) ↔ (𝑌 𝑈) = (𝑏 𝑣)))
7572, 74anbi12d 632 . . . . . . . . 9 (𝑢 = 𝑈 → (((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))
7675anbi2d 630 . . . . . . . 8 (𝑢 = 𝑈 → ((((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))))
7776anbi2d 630 . . . . . . 7 (𝑢 = 𝑈 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))))
78 oveq2 7439 . . . . . . . 8 (𝑢 = 𝑈 → (𝑍 𝑢) = (𝑍 𝑈))
7978eqeq1d 2739 . . . . . . 7 (𝑢 = 𝑈 → ((𝑍 𝑢) = (𝑐 𝑣) ↔ (𝑍 𝑈) = (𝑐 𝑣)))
8077, 79imbi12d 344 . . . . . 6 (𝑢 = 𝑈 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
81802ralbidv 3221 . . . . 5 (𝑢 = 𝑈 → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)) ↔ ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
82 oveq1 7438 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎𝐼𝑐) = (𝐴𝐼𝑐))
8382eleq2d 2827 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑏 ∈ (𝑎𝐼𝑐) ↔ 𝑏 ∈ (𝐴𝐼𝑐)))
84833anbi3d 1444 . . . . . . . 8 (𝑎 = 𝐴 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐))))
85 oveq1 7438 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎 𝑏) = (𝐴 𝑏))
8685eqeq2d 2748 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑋 𝑌) = (𝑎 𝑏) ↔ (𝑋 𝑌) = (𝐴 𝑏)))
8786anbi1d 631 . . . . . . . . 9 (𝑎 = 𝐴 → (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐))))
88 oveq1 7438 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎 𝑣) = (𝐴 𝑣))
8988eqeq2d 2748 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑋 𝑈) = (𝑎 𝑣) ↔ (𝑋 𝑈) = (𝐴 𝑣)))
9089anbi1d 631 . . . . . . . . 9 (𝑎 = 𝐴 → (((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))
9187, 90anbi12d 632 . . . . . . . 8 (𝑎 = 𝐴 → ((((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))))
9284, 91anbi12d 632 . . . . . . 7 (𝑎 = 𝐴 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))))
9392imbi1d 341 . . . . . 6 (𝑎 = 𝐴 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
94932ralbidv 3221 . . . . 5 (𝑎 = 𝐴 → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
95 eleq1 2829 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑏 ∈ (𝐴𝐼𝑐) ↔ 𝐵 ∈ (𝐴𝐼𝑐)))
96953anbi3d 1444 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐))))
97 oveq2 7439 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴 𝑏) = (𝐴 𝐵))
9897eqeq2d 2748 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑋 𝑌) = (𝐴 𝑏) ↔ (𝑋 𝑌) = (𝐴 𝐵)))
99 oveq1 7438 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏 𝑐) = (𝐵 𝑐))
10099eqeq2d 2748 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑌 𝑍) = (𝑏 𝑐) ↔ (𝑌 𝑍) = (𝐵 𝑐)))
10198, 100anbi12d 632 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐))))
102 oveq1 7438 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏 𝑣) = (𝐵 𝑣))
103102eqeq2d 2748 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑌 𝑈) = (𝑏 𝑣) ↔ (𝑌 𝑈) = (𝐵 𝑣)))
104103anbi2d 630 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))))
105101, 104anbi12d 632 . . . . . . . 8 (𝑏 = 𝐵 → ((((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))))
10696, 105anbi12d 632 . . . . . . 7 (𝑏 = 𝐵 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))))))
107106imbi1d 341 . . . . . 6 (𝑏 = 𝐵 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
1081072ralbidv 3221 . . . . 5 (𝑏 = 𝐵 → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
10981, 94, 108rspc3v 3638 . . . 4 ((𝑈𝑃𝐴𝑃𝐵𝑃) → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)) → ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
11068, 69, 70, 109syl3anc 1373 . . 3 (𝜑 → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)) → ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
11167, 110mpd 15 . 2 (𝜑 → ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)))
112 axtg5seg.9 . . . 4 (𝜑𝑋𝑌)
113 axtg5seg.10 . . . 4 (𝜑𝑌 ∈ (𝑋𝐼𝑍))
114 axtg5seg.11 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
115112, 113, 1143jca 1129 . . 3 (𝜑 → (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)))
116 axtg5seg.12 . . . 4 (𝜑 → (𝑋 𝑌) = (𝐴 𝐵))
117 axtg5seg.13 . . . 4 (𝜑 → (𝑌 𝑍) = (𝐵 𝐶))
118116, 117jca 511 . . 3 (𝜑 → ((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)))
119 axtg5seg.14 . . . 4 (𝜑 → (𝑋 𝑈) = (𝐴 𝑉))
120 axtg5seg.15 . . . 4 (𝜑 → (𝑌 𝑈) = (𝐵 𝑉))
121119, 120jca 511 . . 3 (𝜑 → ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))
122115, 118, 121jca32 515 . 2 (𝜑 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))))
123 axtg5seg.6 . . 3 (𝜑𝐶𝑃)
124 axtg5seg.8 . . 3 (𝜑𝑉𝑃)
125 oveq2 7439 . . . . . . . 8 (𝑐 = 𝐶 → (𝐴𝐼𝑐) = (𝐴𝐼𝐶))
126125eleq2d 2827 . . . . . . 7 (𝑐 = 𝐶 → (𝐵 ∈ (𝐴𝐼𝑐) ↔ 𝐵 ∈ (𝐴𝐼𝐶)))
1271263anbi3d 1444 . . . . . 6 (𝑐 = 𝐶 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶))))
128 oveq2 7439 . . . . . . . . 9 (𝑐 = 𝐶 → (𝐵 𝑐) = (𝐵 𝐶))
129128eqeq2d 2748 . . . . . . . 8 (𝑐 = 𝐶 → ((𝑌 𝑍) = (𝐵 𝑐) ↔ (𝑌 𝑍) = (𝐵 𝐶)))
130129anbi2d 630 . . . . . . 7 (𝑐 = 𝐶 → (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶))))
131130anbi1d 631 . . . . . 6 (𝑐 = 𝐶 → ((((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))) ↔ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))))
132127, 131anbi12d 632 . . . . 5 (𝑐 = 𝐶 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))))))
133 oveq1 7438 . . . . . 6 (𝑐 = 𝐶 → (𝑐 𝑣) = (𝐶 𝑣))
134133eqeq2d 2748 . . . . 5 (𝑐 = 𝐶 → ((𝑍 𝑈) = (𝑐 𝑣) ↔ (𝑍 𝑈) = (𝐶 𝑣)))
135132, 134imbi12d 344 . . . 4 (𝑐 = 𝐶 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝐶 𝑣))))
136 oveq2 7439 . . . . . . . . 9 (𝑣 = 𝑉 → (𝐴 𝑣) = (𝐴 𝑉))
137136eqeq2d 2748 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑋 𝑈) = (𝐴 𝑣) ↔ (𝑋 𝑈) = (𝐴 𝑉)))
138 oveq2 7439 . . . . . . . . 9 (𝑣 = 𝑉 → (𝐵 𝑣) = (𝐵 𝑉))
139138eqeq2d 2748 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑌 𝑈) = (𝐵 𝑣) ↔ (𝑌 𝑈) = (𝐵 𝑉)))
140137, 139anbi12d 632 . . . . . . 7 (𝑣 = 𝑉 → (((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉))))
141140anbi2d 630 . . . . . 6 (𝑣 = 𝑉 → ((((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))) ↔ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))))
142141anbi2d 630 . . . . 5 (𝑣 = 𝑉 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉))))))
143 oveq2 7439 . . . . . 6 (𝑣 = 𝑉 → (𝐶 𝑣) = (𝐶 𝑉))
144143eqeq2d 2748 . . . . 5 (𝑣 = 𝑉 → ((𝑍 𝑈) = (𝐶 𝑣) ↔ (𝑍 𝑈) = (𝐶 𝑉)))
145142, 144imbi12d 344 . . . 4 (𝑣 = 𝑉 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝐶 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))) → (𝑍 𝑈) = (𝐶 𝑉))))
146135, 145rspc2v 3633 . . 3 ((𝐶𝑃𝑉𝑃) → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))) → (𝑍 𝑈) = (𝐶 𝑉))))
147123, 124, 146syl2anc 584 . 2 (𝜑 → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))) → (𝑍 𝑈) = (𝐶 𝑉))))
148111, 122, 147mp2d 49 1 (𝜑 → (𝑍 𝑈) = (𝐶 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1086  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  [wsbc 3788  cdif 3948  cin 3950  {csn 4626  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  TarskiGCcstrkgc 28436  TarskiGBcstrkgb 28437  TarskiGCBcstrkgcb 28438  Itvcitv 28441  LineGclng 28442
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-trkgcb 28458  df-trkg 28461
This theorem is referenced by:  tgcgrextend  28493  tgsegconeq  28494  tgifscgr  28516  tgfscgr  28576  tgbtwnconn1lem2  28581  tgbtwnconn1lem3  28582  miriso  28678  midexlem  28700  ragcgr  28715  footexALT  28726  footexlem1  28727  footexlem2  28728  lmiisolem  28804  f1otrg  28879  tg5segofs  34688
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