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Theorem uhgr3cyclexlem 30083
Description: Lemma for uhgr3cyclex 30084. (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 6896 . . . . . . . . 9 (𝐽 = 𝐾 → (𝐼𝐽) = (𝐼𝐾))
21eqeq2d 2736 . . . . . . . 8 (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼𝐽) ↔ {𝐵, 𝐶} = (𝐼𝐾)))
3 eqeq2 2737 . . . . . . . . . . . 12 ((𝐼𝐾) = {𝐶, 𝐴} → ({𝐵, 𝐶} = (𝐼𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
43eqcoms 2733 . . . . . . . . . . 11 ({𝐶, 𝐴} = (𝐼𝐾) → ({𝐵, 𝐶} = (𝐼𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
5 prcom 4738 . . . . . . . . . . . . . 14 {𝐶, 𝐴} = {𝐴, 𝐶}
65eqeq1i 2730 . . . . . . . . . . . . 13 ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
7 simpl 481 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
8 simpr 483 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
97, 8preq1b 4849 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵𝑉) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
109biimpcd 248 . . . . . . . . . . . . 13 ({𝐴, 𝐶} = {𝐵, 𝐶} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
116, 10sylbi 216 . . . . . . . . . . . 12 ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
1211eqcoms 2733 . . . . . . . . . . 11 ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))
134, 12biimtrdi 252 . . . . . . . . . 10 ({𝐶, 𝐴} = (𝐼𝐾) → ({𝐵, 𝐶} = (𝐼𝐾) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵)))
1413adantl 480 . . . . . . . . 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, 15biimtrdi 252 . . . . . . 7 (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼𝐽) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)) → ((𝐴𝑉𝐵𝑉) → 𝐴 = 𝐵))))
1716adantld 489 . . . . . 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 417 . . . 4 (((𝐴𝑉𝐵𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → (𝐽 = 𝐾𝐴 = 𝐵))
2019necon3d 2950 . . 3 (((𝐴𝑉𝐵𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → (𝐴𝐵𝐽𝐾))
2120impancom 450 . 2 (((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾))) → 𝐽𝐾))
2221imp 405 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 205  wa 394   = wceq 1533  wcel 2098  wne 2929  {cpr 4632  dom cdm 5678  cfv 6549  Vtxcvtx 28901  iEdgciedg 28902  Edgcedg 28952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557
This theorem is referenced by:  uhgr3cyclex  30084
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