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Theorem uhgr3cyclexlem 27964
Description: Lemma for uhgr3cyclex 27965. (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 6652 . . . . . . . . 9 (𝐽 = 𝐾 → (𝐼𝐽) = (𝐼𝐾))
21eqeq2d 2833 . . . . . . . 8 (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼𝐽) ↔ {𝐵, 𝐶} = (𝐼𝐾)))
3 eqeq2 2834 . . . . . . . . . . . 12 ((𝐼𝐾) = {𝐶, 𝐴} → ({𝐵, 𝐶} = (𝐼𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
43eqcoms 2830 . . . . . . . . . . 11 ({𝐶, 𝐴} = (𝐼𝐾) → ({𝐵, 𝐶} = (𝐼𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
5 prcom 4642 . . . . . . . . . . . . . 14 {𝐶, 𝐴} = {𝐴, 𝐶}
65eqeq1i 2827 . . . . . . . . . . . . 13 ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
7 simpl 486 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
8 simpr 488 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
97, 8preq1b 4750 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵𝑉) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
109biimpcd 252 . . . . . . . . . . . . 13 ({𝐴, 𝐶} = {𝐵, 𝐶} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
116, 10sylbi 220 . . . . . . . . . . . 12 ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
1211eqcoms 2830 . . . . . . . . . . 11 ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
134, 12syl6bi 256 . . . . . . . . . 10 ({𝐶, 𝐴} = (𝐼𝐾) → ({𝐵, 𝐶} = (𝐼𝐾) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵)))
1413adantl 485 . . . . . . . . 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 256 . . . . . . 7 (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼𝐽) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))))
1716adantld 494 . . . . . 6 (𝐽 = 𝐾 → ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))))
1817com14 96 . . . . 5 ((𝐴𝑉𝐵𝑉) → ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → (𝐽 = 𝐾𝐴 = 𝐵))))
1918imp32 422 . . . 4 (((𝐴𝑉𝐵𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → (𝐽 = 𝐾𝐴 = 𝐵))
2019necon3d 3032 . . 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 410 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 209  wa 399   = wceq 1538  wcel 2114  wne 3011  {cpr 4541  dom cdm 5532  cfv 6334  Vtxcvtx 26787  iEdgciedg 26788  Edgcedg 26838
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 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342
This theorem is referenced by:  uhgr3cyclex  27965
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