Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axtgupdim2ALTV Structured version   Visualization version   GIF version

Theorem axtgupdim2ALTV 32064
 Description: Alternate version of axtgupdim2 26275. (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 1125 . 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 32062 . . . . . 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 221 . . . . 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 7143 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑢) = (𝑋 𝑢))
17 oveq1 7143 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑣) = (𝑋 𝑣))
1816, 17eqeq12d 2814 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑥 𝑢) = (𝑥 𝑣) ↔ (𝑋 𝑢) = (𝑋 𝑣)))
19183anbi1d 1437 . . . . . . . . 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 632 . . . . . . . 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 7143 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑦) = (𝑋𝐼𝑦))
2221eleq2d 2875 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑦)))
23 eleq1 2877 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑦)))
24 oveq1 7143 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2524eleq2d 2875 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
2622, 23, 253orbi123d 1432 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
2720, 26imbi12d 348 . . . . . . 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 3164 . . . . . 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 7143 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 𝑢) = (𝑌 𝑢))
30 oveq1 7143 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 𝑣) = (𝑌 𝑣))
3129, 30eqeq12d 2814 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝑦 𝑢) = (𝑦 𝑣) ↔ (𝑌 𝑢) = (𝑌 𝑣)))
32313anbi2d 1438 . . . . . . . . 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 632 . . . . . . . 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 7144 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑋𝐼𝑦) = (𝑋𝐼𝑌))
3534eleq2d 2875 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑧 ∈ (𝑋𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
36 oveq2 7144 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
3736eleq2d 2875 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
38 eleq1 2877 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
3935, 37, 383orbi123d 1432 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
4033, 39imbi12d 348 . . . . . . 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 3164 . . . . . 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 7143 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑢) = (𝑍 𝑢))
43 oveq1 7143 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑣) = (𝑍 𝑣))
4442, 43eqeq12d 2814 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑧 𝑢) = (𝑧 𝑣) ↔ (𝑍 𝑢) = (𝑍 𝑣)))
45443anbi3d 1439 . . . . . . . . 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 632 . . . . . . . 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 2877 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
48 oveq1 7143 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
4948eleq2d 2875 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
50 oveq2 7144 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5150eleq2d 2875 . . . . . . . . 9 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
5247, 49, 513orbi123d 1432 . . . . . . . 8 (𝑧 = 𝑍 → ((𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
5346, 52imbi12d 348 . . . . . . 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 3164 . . . . . 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 3584 . . . . 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 1368 . . . 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 7144 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑋 𝑢) = (𝑋 𝑈))
6160eqeq1d 2800 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑋 𝑢) = (𝑋 𝑣) ↔ (𝑋 𝑈) = (𝑋 𝑣)))
62 oveq2 7144 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑌 𝑢) = (𝑌 𝑈))
6362eqeq1d 2800 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑌 𝑢) = (𝑌 𝑣) ↔ (𝑌 𝑈) = (𝑌 𝑣)))
64 oveq2 7144 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑍 𝑢) = (𝑍 𝑈))
6564eqeq1d 2800 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑍 𝑢) = (𝑍 𝑣) ↔ (𝑍 𝑈) = (𝑍 𝑣)))
6661, 63, 653anbi123d 1433 . . . . . . 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 3049 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝑣𝑈𝑣))
6866, 67anbi12d 633 . . . . . 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 345 . . . . 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 7144 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑋 𝑣) = (𝑋 𝑉))
7170eqeq2d 2809 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑋 𝑈) = (𝑋 𝑣) ↔ (𝑋 𝑈) = (𝑋 𝑉)))
72 oveq2 7144 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑌 𝑣) = (𝑌 𝑉))
7372eqeq2d 2809 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑌 𝑈) = (𝑌 𝑣) ↔ (𝑌 𝑈) = (𝑌 𝑉)))
74 oveq2 7144 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑍 𝑣) = (𝑍 𝑉))
7574eqeq2d 2809 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑍 𝑈) = (𝑍 𝑣) ↔ (𝑍 𝑈) = (𝑍 𝑉)))
7671, 73, 753anbi123d 1433 . . . . . . 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 3050 . . . . . . 7 (𝑣 = 𝑉 → (𝑈𝑣𝑈𝑉))
7876, 77anbi12d 633 . . . . . 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 345 . . . . 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 3581 . . . 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 587 . . 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 698 1 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  distcds 16569  Itvcitv 26240  TarskiG2Dcstrkg2d 32060 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-ov 7139  df-trkg2d 32061 This theorem is referenced by: (None)
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