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Theorem axtg5seg 25081
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 25069 . . . . . . 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 3795 . . . . . . . 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 3794 . . . . . . . 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 3576 . . . . . . 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 3597 . . . . . 6 TarskiG ⊆ TarskiGCB
6 axtrkg.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3565 . . . . 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 25072 . . . . . . 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 478 . . . . . 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 473 . . . . 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 2843 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
19 oveq1 6534 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2019eleq2d 2672 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
2118, 203anbi12d 1391 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
22 oveq1 6534 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
2322eqeq1d 2611 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑎 𝑏) ↔ (𝑋 𝑦) = (𝑎 𝑏)))
2423anbi1d 736 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐))))
25 oveq1 6534 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 𝑢) = (𝑋 𝑢))
2625eqeq1d 2611 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 𝑢) = (𝑎 𝑣) ↔ (𝑋 𝑢) = (𝑎 𝑣)))
2726anbi1d 736 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))))
2824, 27anbi12d 742 . . . . . . . . . . 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 742 . . . . . . . . . 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 329 . . . . . . . . 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 2968 . . . . . . . 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 2971 . . . . . . 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 2971 . . . . . 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 2844 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋𝑦𝑋𝑌))
35 eleq1 2675 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
3634, 353anbi12d 1391 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
37 oveq2 6535 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
3837eqeq1d 2611 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑎 𝑏) ↔ (𝑋 𝑌) = (𝑎 𝑏)))
39 oveq1 6534 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
4039eqeq1d 2611 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑦 𝑧) = (𝑏 𝑐) ↔ (𝑌 𝑧) = (𝑏 𝑐)))
4138, 40anbi12d 742 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐))))
42 oveq1 6534 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑦 𝑢) = (𝑌 𝑢))
4342eqeq1d 2611 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑦 𝑢) = (𝑏 𝑣) ↔ (𝑌 𝑢) = (𝑏 𝑣)))
4443anbi2d 735 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))))
4541, 44anbi12d 742 . . . . . . . . . . 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 742 . . . . . . . . . 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 329 . . . . . . . . 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 2968 . . . . . . . 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 2971 . . . . . . 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 2971 . . . . . 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 6535 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5251eleq2d 2672 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
53523anbi2d 1395 . . . . . . . . . . 11 (𝑧 = 𝑍 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
54 oveq2 6535 . . . . . . . . . . . . . 14 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
5554eqeq1d 2611 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → ((𝑌 𝑧) = (𝑏 𝑐) ↔ (𝑌 𝑍) = (𝑏 𝑐)))
5655anbi2d 735 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐))))
5756anbi1d 736 . . . . . . . . . . 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 742 . . . . . . . . . 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 6534 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑢) = (𝑍 𝑢))
6059eqeq1d 2611 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑧 𝑢) = (𝑐 𝑣) ↔ (𝑍 𝑢) = (𝑐 𝑣)))
6158, 60imbi12d 332 . . . . . . . . 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 2968 . . . . . . . 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 2971 . . . . . . 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 2971 . . . . . 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 3295 . . . . 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 1317 . . . 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 6535 . . . . . . . . . . 11 (𝑢 = 𝑈 → (𝑋 𝑢) = (𝑋 𝑈))
7271eqeq1d 2611 . . . . . . . . . 10 (𝑢 = 𝑈 → ((𝑋 𝑢) = (𝑎 𝑣) ↔ (𝑋 𝑈) = (𝑎 𝑣)))
73 oveq2 6535 . . . . . . . . . . 11 (𝑢 = 𝑈 → (𝑌 𝑢) = (𝑌 𝑈))
7473eqeq1d 2611 . . . . . . . . . 10 (𝑢 = 𝑈 → ((𝑌 𝑢) = (𝑏 𝑣) ↔ (𝑌 𝑈) = (𝑏 𝑣)))
7572, 74anbi12d 742 . . . . . . . . 9 (𝑢 = 𝑈 → (((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))
7675anbi2d 735 . . . . . . . 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 735 . . . . . . 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 6535 . . . . . . . 8 (𝑢 = 𝑈 → (𝑍 𝑢) = (𝑍 𝑈))
7978eqeq1d 2611 . . . . . . 7 (𝑢 = 𝑈 → ((𝑍 𝑢) = (𝑐 𝑣) ↔ (𝑍 𝑈) = (𝑐 𝑣)))
8077, 79imbi12d 332 . . . . . 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 2971 . . . . 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 6534 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎𝐼𝑐) = (𝐴𝐼𝑐))
8382eleq2d 2672 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑏 ∈ (𝑎𝐼𝑐) ↔ 𝑏 ∈ (𝐴𝐼𝑐)))
84833anbi3d 1396 . . . . . . . 8 (𝑎 = 𝐴 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐))))
85 oveq1 6534 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎 𝑏) = (𝐴 𝑏))
8685eqeq2d 2619 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑋 𝑌) = (𝑎 𝑏) ↔ (𝑋 𝑌) = (𝐴 𝑏)))
8786anbi1d 736 . . . . . . . . 9 (𝑎 = 𝐴 → (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐))))
88 oveq1 6534 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎 𝑣) = (𝐴 𝑣))
8988eqeq2d 2619 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑋 𝑈) = (𝑎 𝑣) ↔ (𝑋 𝑈) = (𝐴 𝑣)))
9089anbi1d 736 . . . . . . . . 9 (𝑎 = 𝐴 → (((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))
9187, 90anbi12d 742 . . . . . . . 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 742 . . . . . . 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 329 . . . . . 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 2971 . . . . 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 2675 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑏 ∈ (𝐴𝐼𝑐) ↔ 𝐵 ∈ (𝐴𝐼𝑐)))
96953anbi3d 1396 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐))))
97 oveq2 6535 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴 𝑏) = (𝐴 𝐵))
9897eqeq2d 2619 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑋 𝑌) = (𝐴 𝑏) ↔ (𝑋 𝑌) = (𝐴 𝐵)))
99 oveq1 6534 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏 𝑐) = (𝐵 𝑐))
10099eqeq2d 2619 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑌 𝑍) = (𝑏 𝑐) ↔ (𝑌 𝑍) = (𝐵 𝑐)))
10198, 100anbi12d 742 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐))))
102 oveq1 6534 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏 𝑣) = (𝐵 𝑣))
103102eqeq2d 2619 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑌 𝑈) = (𝑏 𝑣) ↔ (𝑌 𝑈) = (𝐵 𝑣)))
104103anbi2d 735 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))))
105101, 104anbi12d 742 . . . . . . . 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 742 . . . . . . 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 329 . . . . . 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 2971 . . . . 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 3295 . . . 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 1317 . . 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 1234 . . 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 552 . . 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 552 . . 3 (𝜑 → ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))
122115, 118, 121jca32 555 . 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 6535 . . . . . . . 8 (𝑐 = 𝐶 → (𝐴𝐼𝑐) = (𝐴𝐼𝐶))
126125eleq2d 2672 . . . . . . 7 (𝑐 = 𝐶 → (𝐵 ∈ (𝐴𝐼𝑐) ↔ 𝐵 ∈ (𝐴𝐼𝐶)))
1271263anbi3d 1396 . . . . . 6 (𝑐 = 𝐶 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶))))
128 oveq2 6535 . . . . . . . . 9 (𝑐 = 𝐶 → (𝐵 𝑐) = (𝐵 𝐶))
129128eqeq2d 2619 . . . . . . . 8 (𝑐 = 𝐶 → ((𝑌 𝑍) = (𝐵 𝑐) ↔ (𝑌 𝑍) = (𝐵 𝐶)))
130129anbi2d 735 . . . . . . 7 (𝑐 = 𝐶 → (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶))))
131130anbi1d 736 . . . . . 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 742 . . . . 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 6534 . . . . . 6 (𝑐 = 𝐶 → (𝑐 𝑣) = (𝐶 𝑣))
134133eqeq2d 2619 . . . . 5 (𝑐 = 𝐶 → ((𝑍 𝑈) = (𝑐 𝑣) ↔ (𝑍 𝑈) = (𝐶 𝑣)))
135132, 134imbi12d 332 . . . 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 6535 . . . . . . . . 9 (𝑣 = 𝑉 → (𝐴 𝑣) = (𝐴 𝑉))
137136eqeq2d 2619 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑋 𝑈) = (𝐴 𝑣) ↔ (𝑋 𝑈) = (𝐴 𝑉)))
138 oveq2 6535 . . . . . . . . 9 (𝑣 = 𝑉 → (𝐵 𝑣) = (𝐵 𝑉))
139138eqeq2d 2619 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑌 𝑈) = (𝐵 𝑣) ↔ (𝑌 𝑈) = (𝐵 𝑉)))
140137, 139anbi12d 742 . . . . . . 7 (𝑣 = 𝑉 → (((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉))))
141140anbi2d 735 . . . . . 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 735 . . . . 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 6535 . . . . . 6 (𝑣 = 𝑉 → (𝐶 𝑣) = (𝐶 𝑉))
144143eqeq2d 2619 . . . . 5 (𝑣 = 𝑉 → ((𝑍 𝑈) = (𝐶 𝑣) ↔ (𝑍 𝑈) = (𝐶 𝑉)))
145142, 144imbi12d 332 . . . 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 3292 . . 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 690 . 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 46 1 (𝜑 → (𝑍 𝑈) = (𝐶 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3o 1029  w3a 1030   = wceq 1474  wcel 1976  {cab 2595  wne 2779  wral 2895  wrex 2896  {crab 2899  Vcvv 3172  [wsbc 3401  cdif 3536  cin 3538  {csn 4124  cfv 5790  (class class class)co 6527  cmpt2 6529  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  TarskiGCcstrkgc 25047  TarskiGBcstrkgb 25048  TarskiGCBcstrkgcb 25049  Itvcitv 25052  LineGclng 25053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-trkgcb 25066  df-trkg 25069
This theorem is referenced by:  tgcgrextend  25097  tgsegconeq  25098  tgifscgr  25121  tgfscgr  25181  tgbtwnconn1lem2  25186  tgbtwnconn1lem3  25187  miriso  25283  midexlem  25305  ragcgr  25320  footex  25331  lmiisolem  25406  f1otrg  25469  tg5segofs  29810
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