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Theorem isplig 26720
 Description: The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)
Hypothesis
Ref Expression
isplig.1 𝑃 = 𝐿
Assertion
Ref Expression
isplig (𝐿𝐴 → (𝐿 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐿𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
Distinct variable groups:   𝐿,𝑎,𝑏,𝑐,𝑙   𝑃,𝑎,𝑏,𝑐
Allowed substitution hints:   𝐴(𝑎,𝑏,𝑐,𝑙)   𝑃(𝑙)

Proof of Theorem isplig
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 unieq 4380 . . . . 5 (𝑥 = 𝐿 𝑥 = 𝐿)
2 isplig.1 . . . . 5 𝑃 = 𝐿
31, 2syl6eqr 2662 . . . 4 (𝑥 = 𝐿 𝑥 = 𝑃)
4 reueq1 3117 . . . . . 6 (𝑥 = 𝐿 → (∃!𝑙𝑥 (𝑎𝑙𝑏𝑙) ↔ ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙)))
54imbi2d 329 . . . . 5 (𝑥 = 𝐿 → ((𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙))))
63, 5raleqbidv 3129 . . . 4 (𝑥 = 𝐿 → (∀𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ ∀𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙))))
73, 6raleqbidv 3129 . . 3 (𝑥 = 𝐿 → (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ ∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙))))
83rexeqdv 3122 . . . . 5 (𝑥 = 𝐿 → (∃𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
93, 8rexeqbidv 3130 . . . 4 (𝑥 = 𝐿 → (∃𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
109raleqbi1dv 3123 . . 3 (𝑥 = 𝐿 → (∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∀𝑙𝐿𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
11 raleq 3115 . . . . . 6 (𝑥 = 𝐿 → (∀𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∀𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
123, 11rexeqbidv 3130 . . . . 5 (𝑥 = 𝐿 → (∃𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
133, 12rexeqbidv 3130 . . . 4 (𝑥 = 𝐿 → (∃𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
143, 13rexeqbidv 3130 . . 3 (𝑥 = 𝐿 → (∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
157, 10, 143anbi123d 1391 . 2 (𝑥 = 𝐿 → ((∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)) ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐿𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
16 df-plig 26719 . 2 Plig = {𝑥 ∣ (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))}
1715, 16elab2g 3322 1 (𝐿𝐴 → (𝐿 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐿𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∪ cuni 4372  Pligcplig 26718 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-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-v 3175  df-uni 4373  df-plig 26719 This theorem is referenced by:  tncp  26721
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