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Theorem axtgupdim2ALTV 34662
Description: Alternate version of axtgupdim2 28494. (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 1127 . 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 34660 . . . . . 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 774 . . . 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 7438 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑢) = (𝑋 𝑢))
17 oveq1 7438 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑣) = (𝑋 𝑣))
1816, 17eqeq12d 2751 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑥 𝑢) = (𝑥 𝑣) ↔ (𝑋 𝑢) = (𝑋 𝑣)))
19183anbi1d 1439 . . . . . . . . 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 7438 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑦) = (𝑋𝐼𝑦))
2221eleq2d 2825 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑦)))
23 eleq1 2827 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑦)))
24 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2524eleq2d 2825 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
2622, 23, 253orbi123d 1434 . . . . . . . 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 3219 . . . . . 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 7438 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 𝑢) = (𝑌 𝑢))
30 oveq1 7438 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 𝑣) = (𝑌 𝑣))
3129, 30eqeq12d 2751 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝑦 𝑢) = (𝑦 𝑣) ↔ (𝑌 𝑢) = (𝑌 𝑣)))
32313anbi2d 1440 . . . . . . . . 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 7439 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑋𝐼𝑦) = (𝑋𝐼𝑌))
3534eleq2d 2825 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑧 ∈ (𝑋𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
36 oveq2 7439 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
3736eleq2d 2825 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
38 eleq1 2827 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
3935, 37, 383orbi123d 1434 . . . . . . . 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 3219 . . . . . 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 7438 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑢) = (𝑍 𝑢))
43 oveq1 7438 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑣) = (𝑍 𝑣))
4442, 43eqeq12d 2751 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑧 𝑢) = (𝑧 𝑣) ↔ (𝑍 𝑢) = (𝑍 𝑣)))
45443anbi3d 1441 . . . . . . . . 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 2827 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
48 oveq1 7438 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
4948eleq2d 2825 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
50 oveq2 7439 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5150eleq2d 2825 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
5247, 49, 513orbi123d 1434 . . . . . . . 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 3219 . . . . . 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 3638 . . . . 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 1370 . . . 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 7439 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑋 𝑢) = (𝑋 𝑈))
6160eqeq1d 2737 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑋 𝑢) = (𝑋 𝑣) ↔ (𝑋 𝑈) = (𝑋 𝑣)))
62 oveq2 7439 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑌 𝑢) = (𝑌 𝑈))
6362eqeq1d 2737 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑌 𝑢) = (𝑌 𝑣) ↔ (𝑌 𝑈) = (𝑌 𝑣)))
64 oveq2 7439 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑍 𝑢) = (𝑍 𝑈))
6564eqeq1d 2737 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑍 𝑢) = (𝑍 𝑣) ↔ (𝑍 𝑈) = (𝑍 𝑣)))
6661, 63, 653anbi123d 1435 . . . . . . 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 3001 . . . . . . 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 7439 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑋 𝑣) = (𝑋 𝑉))
7170eqeq2d 2746 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑋 𝑈) = (𝑋 𝑣) ↔ (𝑋 𝑈) = (𝑋 𝑉)))
72 oveq2 7439 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑌 𝑣) = (𝑌 𝑉))
7372eqeq2d 2746 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑌 𝑈) = (𝑌 𝑣) ↔ (𝑌 𝑈) = (𝑌 𝑉)))
74 oveq2 7439 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑍 𝑣) = (𝑍 𝑉))
7574eqeq2d 2746 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑍 𝑈) = (𝑍 𝑣) ↔ (𝑍 𝑈) = (𝑍 𝑉)))
7671, 73, 753anbi123d 1435 . . . . . . 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 3002 . . . . . . 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 3633 . . . 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 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  Itvcitv 28456  TarskiG2Dcstrkg2d 34658
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-trkg2d 34659
This theorem is referenced by: (None)
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