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Theorem isplig 26486
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 4374 . . . . 5 (𝑥 = 𝐿 𝑥 = 𝐿)
2 isplig.1 . . . . 5 𝑃 = 𝐿
31, 2syl6eqr 2661 . . . 4 (𝑥 = 𝐿 𝑥 = 𝑃)
4 reueq1 3116 . . . . . 6 (𝑥 = 𝐿 → (∃!𝑙𝑥 (𝑎𝑙𝑏𝑙) ↔ ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙)))
54imbi2d 328 . . . . 5 (𝑥 = 𝐿 → ((𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙))))
63, 5raleqbidv 3128 . . . 4 (𝑥 = 𝐿 → (∀𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ ∀𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙))))
73, 6raleqbidv 3128 . . 3 (𝑥 = 𝐿 → (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ ∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙))))
83rexeqdv 3121 . . . . 5 (𝑥 = 𝐿 → (∃𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
93, 8rexeqbidv 3129 . . . 4 (𝑥 = 𝐿 → (∃𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
109raleqbi1dv 3122 . . 3 (𝑥 = 𝐿 → (∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∀𝑙𝐿𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
11 raleq 3114 . . . . . 6 (𝑥 = 𝐿 → (∀𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∀𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
123, 11rexeqbidv 3129 . . . . 5 (𝑥 = 𝐿 → (∃𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
133, 12rexeqbidv 3129 . . . 4 (𝑥 = 𝐿 → (∃𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
143, 13rexeqbidv 3129 . . 3 (𝑥 = 𝐿 → (∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
157, 10, 143anbi123d 1390 . 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 26485 . 2 Plig = {𝑥 ∣ (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))}
1715, 16elab2g 3321 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  ∃!wreu 2897   cuni 4366  Pligcplig 26484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-reu 2902  df-v 3174  df-uni 4367  df-plig 26485
This theorem is referenced by:  tncp  26487
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