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Theorem isplig 27664
Description: The predicate "is a planar incidence geometry" for sets. (Contributed by FL, 2-Aug-2009.)
Hypothesis
Ref Expression
isplig.1 𝑃 = 𝐺
Assertion
Ref Expression
isplig (𝐺𝐴 → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑙,𝐺   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 4580 . . . . 5 (𝑥 = 𝐺 𝑥 = 𝐺)
2 isplig.1 . . . . 5 𝑃 = 𝐺
31, 2syl6eqr 2822 . . . 4 (𝑥 = 𝐺 𝑥 = 𝑃)
4 reueq1 3288 . . . . . 6 (𝑥 = 𝐺 → (∃!𝑙𝑥 (𝑎𝑙𝑏𝑙) ↔ ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)))
54imbi2d 329 . . . . 5 (𝑥 = 𝐺 → ((𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙))))
63, 5raleqbidv 3300 . . . 4 (𝑥 = 𝐺 → (∀𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ ∀𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙))))
73, 6raleqbidv 3300 . . 3 (𝑥 = 𝐺 → (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ ∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙))))
83rexeqdv 3293 . . . . 5 (𝑥 = 𝐺 → (∃𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
93, 8rexeqbidv 3301 . . . 4 (𝑥 = 𝐺 → (∃𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
109raleqbi1dv 3294 . . 3 (𝑥 = 𝐺 → (∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
11 raleq 3286 . . . . . 6 (𝑥 = 𝐺 → (∀𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∀𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
123, 11rexeqbidv 3301 . . . . 5 (𝑥 = 𝐺 → (∃𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
133, 12rexeqbidv 3301 . . . 4 (𝑥 = 𝐺 → (∃𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
143, 13rexeqbidv 3301 . . 3 (𝑥 = 𝐺 → (∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
157, 10, 143anbi123d 1546 . 2 (𝑥 = 𝐺 → ((∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)) ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
16 df-plig 27663 . 2 Plig = {𝑥 ∣ (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))}
1715, 16elab2g 3502 1 (𝐺𝐴 → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  ∃!wreu 3062   cuni 4572  Pligcplig 27662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-reu 3067  df-v 3351  df-uni 4573  df-plig 27663
This theorem is referenced by:  ispligb  27665  tncp  27666  l2p  27667  eulplig  27673
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