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Theorem uhgr3cyclexlem 41348
 Description: Lemma for uhgr3cyclex 41349. (Contributed by AV, 12-Feb-2021.)
Hypotheses
Ref Expression
uhgr3cyclex.v 𝑉 = (Vtx‘𝐺)
uhgr3cyclex.e 𝐸 = (Edg‘𝐺)
uhgr3cyclex.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
uhgr3cyclexlem ((((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → 𝐽𝐾)

Proof of Theorem uhgr3cyclexlem
StepHypRef Expression
1 fveq2 6103 . . . . . . . . 9 (𝐽 = 𝐾 → (𝐼𝐽) = (𝐼𝐾))
21eqeq2d 2620 . . . . . . . 8 (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼𝐽) ↔ {𝐵, 𝐶} = (𝐼𝐾)))
3 eqeq2 2621 . . . . . . . . . . . 12 ((𝐼𝐾) = {𝐶, 𝐴} → ({𝐵, 𝐶} = (𝐼𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
43eqcoms 2618 . . . . . . . . . . 11 ({𝐶, 𝐴} = (𝐼𝐾) → ({𝐵, 𝐶} = (𝐼𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
5 prcom 4211 . . . . . . . . . . . . . 14 {𝐶, 𝐴} = {𝐴, 𝐶}
65eqeq1i 2615 . . . . . . . . . . . . 13 ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
7 simpl 472 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
8 simpr 476 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
97, 8preq1b 4317 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵𝑉) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
109biimpcd 238 . . . . . . . . . . . . 13 ({𝐴, 𝐶} = {𝐵, 𝐶} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
116, 10sylbi 206 . . . . . . . . . . . 12 ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
1211eqcoms 2618 . . . . . . . . . . 11 ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
134, 12syl6bi 242 . . . . . . . . . 10 ({𝐶, 𝐴} = (𝐼𝐾) → ({𝐵, 𝐶} = (𝐼𝐾) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵)))
1413adantl 481 . . . . . . . . 9 ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ({𝐵, 𝐶} = (𝐼𝐾) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵)))
1514com12 32 . . . . . . . 8 ({𝐵, 𝐶} = (𝐼𝐾) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵)))
162, 15syl6bi 242 . . . . . . 7 (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼𝐽) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))))
1716adantld 482 . . . . . 6 (𝐽 = 𝐾 → ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))))
1817com14 94 . . . . 5 ((𝐴𝑉𝐵𝑉) → ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → (𝐽 = 𝐾𝐴 = 𝐵))))
1918imp32 448 . . . 4 (((𝐴𝑉𝐵𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → (𝐽 = 𝐾𝐴 = 𝐵))
2019necon3d 2803 . . 3 (((𝐴𝑉𝐵𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → (𝐴𝐵𝐽𝐾))
2120impancom 455 . 2 (((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾))) → 𝐽𝐾))
2221imp 444 1 ((((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → 𝐽𝐾)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127  dom cdm 5038  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  uhgr3cyclex  41349
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