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Theorem axtgupdim2ALTV 34684
Description: Alternate version of axtgupdim2 28480. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.)
Hypotheses
Ref Expression
istrkg2d.p 𝑃 = (Base‘𝐺)
istrkg2d.d = (dist‘𝐺)
istrkg2d.i 𝐼 = (Itv‘𝐺)
axtgupdim2ALTV.x (𝜑𝑋𝑃)
axtgupdim2ALTV.y (𝜑𝑌𝑃)
axtgupdim2ALTV.z (𝜑𝑍𝑃)
axtgupdim2ALTV.u (𝜑𝑈𝑃)
axtgupdim2ALTV.v (𝜑𝑉𝑃)
axtgupdim2ALTV.0 (𝜑𝑈𝑉)
axtgupdim2ALTV.1 (𝜑 → (𝑋 𝑈) = (𝑋 𝑉))
axtgupdim2ALTV.2 (𝜑 → (𝑌 𝑈) = (𝑌 𝑉))
axtgupdim2ALTV.3 (𝜑 → (𝑍 𝑈) = (𝑍 𝑉))
axtgupdim2ALTV.g (𝜑𝐺 ∈ TarskiG2D)
Assertion
Ref Expression
axtgupdim2ALTV (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))

Proof of Theorem axtgupdim2ALTV
Dummy variables 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axtgupdim2ALTV.1 . . 3 (𝜑 → (𝑋 𝑈) = (𝑋 𝑉))
2 axtgupdim2ALTV.2 . . 3 (𝜑 → (𝑌 𝑈) = (𝑌 𝑉))
3 axtgupdim2ALTV.3 . . 3 (𝜑 → (𝑍 𝑈) = (𝑍 𝑉))
41, 2, 33jca 1128 . 2 (𝜑 → ((𝑋 𝑈) = (𝑋 𝑉) ∧ (𝑌 𝑈) = (𝑌 𝑉) ∧ (𝑍 𝑈) = (𝑍 𝑉)))
5 axtgupdim2ALTV.0 . 2 (𝜑𝑈𝑉)
6 axtgupdim2ALTV.g . . . . . 6 (𝜑𝐺 ∈ TarskiG2D)
7 istrkg2d.p . . . . . . 7 𝑃 = (Base‘𝐺)
8 istrkg2d.d . . . . . . 7 = (dist‘𝐺)
9 istrkg2d.i . . . . . . 7 𝐼 = (Itv‘𝐺)
107, 8, 9istrkg2d 34682 . . . . . 6 (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
116, 10sylib 218 . . . . 5 (𝜑 → (𝐺 ∈ V ∧ (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
1211simprrd 773 . . . 4 (𝜑 → ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
13 axtgupdim2ALTV.x . . . . 5 (𝜑𝑋𝑃)
14 axtgupdim2ALTV.y . . . . 5 (𝜑𝑌𝑃)
15 axtgupdim2ALTV.z . . . . 5 (𝜑𝑍𝑃)
16 oveq1 7439 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑢) = (𝑋 𝑢))
17 oveq1 7439 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑣) = (𝑋 𝑣))
1816, 17eqeq12d 2752 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑥 𝑢) = (𝑥 𝑣) ↔ (𝑋 𝑢) = (𝑋 𝑣)))
19183anbi1d 1441 . . . . . . . . 9 (𝑥 = 𝑋 → (((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ↔ ((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣))))
2019anbi1d 631 . . . . . . . 8 (𝑥 = 𝑋 → ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) ↔ (((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣)))
21 oveq1 7439 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑦) = (𝑋𝐼𝑦))
2221eleq2d 2826 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑦)))
23 eleq1 2828 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑦)))
24 oveq1 7439 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2524eleq2d 2826 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
2622, 23, 253orbi123d 1436 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
2720, 26imbi12d 344 . . . . . . 7 (𝑥 = 𝑋 → (((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)))))
28272ralbidv 3220 . . . . . 6 (𝑥 = 𝑋 → (∀𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)))))
29 oveq1 7439 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 𝑢) = (𝑌 𝑢))
30 oveq1 7439 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 𝑣) = (𝑌 𝑣))
3129, 30eqeq12d 2752 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝑦 𝑢) = (𝑦 𝑣) ↔ (𝑌 𝑢) = (𝑌 𝑣)))
32313anbi2d 1442 . . . . . . . . 9 (𝑦 = 𝑌 → (((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ↔ ((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣))))
3332anbi1d 631 . . . . . . . 8 (𝑦 = 𝑌 → ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) ↔ (((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣)))
34 oveq2 7440 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑋𝐼𝑦) = (𝑋𝐼𝑌))
3534eleq2d 2826 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑧 ∈ (𝑋𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
36 oveq2 7440 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
3736eleq2d 2826 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
38 eleq1 2828 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
3935, 37, 383orbi123d 1436 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
4033, 39imbi12d 344 . . . . . . 7 (𝑦 = 𝑌 → (((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))) ↔ ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)))))
41402ralbidv 3220 . . . . . 6 (𝑦 = 𝑌 → (∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))) ↔ ∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)))))
42 oveq1 7439 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑢) = (𝑍 𝑢))
43 oveq1 7439 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑣) = (𝑍 𝑣))
4442, 43eqeq12d 2752 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑧 𝑢) = (𝑧 𝑣) ↔ (𝑍 𝑢) = (𝑍 𝑣)))
45443anbi3d 1443 . . . . . . . . 9 (𝑧 = 𝑍 → (((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ↔ ((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣))))
4645anbi1d 631 . . . . . . . 8 (𝑧 = 𝑍 → ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) ↔ (((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣)))
47 eleq1 2828 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
48 oveq1 7439 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
4948eleq2d 2826 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
50 oveq2 7440 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5150eleq2d 2826 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
5247, 49, 513orbi123d 1436 . . . . . . . 8 (𝑧 = 𝑍 → ((𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
5346, 52imbi12d 344 . . . . . . 7 (𝑧 = 𝑍 → (((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))) ↔ ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
54532ralbidv 3220 . . . . . 6 (𝑧 = 𝑍 → (∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))) ↔ ∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
5528, 41, 54rspc3v 3637 . . . . 5 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) → ∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
5613, 14, 15, 55syl3anc 1372 . . . 4 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) → ∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
5712, 56mpd 15 . . 3 (𝜑 → ∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
58 axtgupdim2ALTV.u . . . 4 (𝜑𝑈𝑃)
59 axtgupdim2ALTV.v . . . 4 (𝜑𝑉𝑃)
60 oveq2 7440 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑋 𝑢) = (𝑋 𝑈))
6160eqeq1d 2738 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑋 𝑢) = (𝑋 𝑣) ↔ (𝑋 𝑈) = (𝑋 𝑣)))
62 oveq2 7440 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑌 𝑢) = (𝑌 𝑈))
6362eqeq1d 2738 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑌 𝑢) = (𝑌 𝑣) ↔ (𝑌 𝑈) = (𝑌 𝑣)))
64 oveq2 7440 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑍 𝑢) = (𝑍 𝑈))
6564eqeq1d 2738 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑍 𝑢) = (𝑍 𝑣) ↔ (𝑍 𝑈) = (𝑍 𝑣)))
6661, 63, 653anbi123d 1437 . . . . . . 7 (𝑢 = 𝑈 → (((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ↔ ((𝑋 𝑈) = (𝑋 𝑣) ∧ (𝑌 𝑈) = (𝑌 𝑣) ∧ (𝑍 𝑈) = (𝑍 𝑣))))
67 neeq1 3002 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝑣𝑈𝑣))
6866, 67anbi12d 632 . . . . . 6 (𝑢 = 𝑈 → ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) ↔ (((𝑋 𝑈) = (𝑋 𝑣) ∧ (𝑌 𝑈) = (𝑌 𝑣) ∧ (𝑍 𝑈) = (𝑍 𝑣)) ∧ 𝑈𝑣)))
6968imbi1d 341 . . . . 5 (𝑢 = 𝑈 → (((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) ↔ ((((𝑋 𝑈) = (𝑋 𝑣) ∧ (𝑌 𝑈) = (𝑌 𝑣) ∧ (𝑍 𝑈) = (𝑍 𝑣)) ∧ 𝑈𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
70 oveq2 7440 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑋 𝑣) = (𝑋 𝑉))
7170eqeq2d 2747 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑋 𝑈) = (𝑋 𝑣) ↔ (𝑋 𝑈) = (𝑋 𝑉)))
72 oveq2 7440 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑌 𝑣) = (𝑌 𝑉))
7372eqeq2d 2747 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑌 𝑈) = (𝑌 𝑣) ↔ (𝑌 𝑈) = (𝑌 𝑉)))
74 oveq2 7440 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑍 𝑣) = (𝑍 𝑉))
7574eqeq2d 2747 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑍 𝑈) = (𝑍 𝑣) ↔ (𝑍 𝑈) = (𝑍 𝑉)))
7671, 73, 753anbi123d 1437 . . . . . . 7 (𝑣 = 𝑉 → (((𝑋 𝑈) = (𝑋 𝑣) ∧ (𝑌 𝑈) = (𝑌 𝑣) ∧ (𝑍 𝑈) = (𝑍 𝑣)) ↔ ((𝑋 𝑈) = (𝑋 𝑉) ∧ (𝑌 𝑈) = (𝑌 𝑉) ∧ (𝑍 𝑈) = (𝑍 𝑉))))
77 neeq2 3003 . . . . . . 7 (𝑣 = 𝑉 → (𝑈𝑣𝑈𝑉))
7876, 77anbi12d 632 . . . . . 6 (𝑣 = 𝑉 → ((((𝑋 𝑈) = (𝑋 𝑣) ∧ (𝑌 𝑈) = (𝑌 𝑣) ∧ (𝑍 𝑈) = (𝑍 𝑣)) ∧ 𝑈𝑣) ↔ (((𝑋 𝑈) = (𝑋 𝑉) ∧ (𝑌 𝑈) = (𝑌 𝑉) ∧ (𝑍 𝑈) = (𝑍 𝑉)) ∧ 𝑈𝑉)))
7978imbi1d 341 . . . . 5 (𝑣 = 𝑉 → (((((𝑋 𝑈) = (𝑋 𝑣) ∧ (𝑌 𝑈) = (𝑌 𝑣) ∧ (𝑍 𝑈) = (𝑍 𝑣)) ∧ 𝑈𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) ↔ ((((𝑋 𝑈) = (𝑋 𝑉) ∧ (𝑌 𝑈) = (𝑌 𝑉) ∧ (𝑍 𝑈) = (𝑍 𝑉)) ∧ 𝑈𝑉) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
8069, 79rspc2v 3632 . . . 4 ((𝑈𝑃𝑉𝑃) → (∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) → ((((𝑋 𝑈) = (𝑋 𝑉) ∧ (𝑌 𝑈) = (𝑌 𝑉) ∧ (𝑍 𝑈) = (𝑍 𝑉)) ∧ 𝑈𝑉) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
8158, 59, 80syl2anc 584 . . 3 (𝜑 → (∀𝑢𝑃𝑣𝑃 ((((𝑋 𝑢) = (𝑋 𝑣) ∧ (𝑌 𝑢) = (𝑌 𝑣) ∧ (𝑍 𝑢) = (𝑍 𝑣)) ∧ 𝑢𝑣) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) → ((((𝑋 𝑈) = (𝑋 𝑉) ∧ (𝑌 𝑈) = (𝑌 𝑉) ∧ (𝑍 𝑈) = (𝑍 𝑉)) ∧ 𝑈𝑉) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
8257, 81mpd 15 . 2 (𝜑 → ((((𝑋 𝑈) = (𝑋 𝑉) ∧ (𝑌 𝑈) = (𝑌 𝑉) ∧ (𝑍 𝑈) = (𝑍 𝑉)) ∧ 𝑈𝑉) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
834, 5, 82mp2and 699 1 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3o 1085  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cfv 6560  (class class class)co 7432  Basecbs 17248  distcds 17307  Itvcitv 28442  TarskiG2Dcstrkg2d 34680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-trkg2d 34681
This theorem is referenced by: (None)
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