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Theorem isplig 29247
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 4874 . . . . 5 (𝑥 = 𝐺 𝑥 = 𝐺)
2 isplig.1 . . . . 5 𝑃 = 𝐺
31, 2eqtr4di 2795 . . . 4 (𝑥 = 𝐺 𝑥 = 𝑃)
4 reueq1 3390 . . . . . 6 (𝑥 = 𝐺 → (∃!𝑙𝑥 (𝑎𝑙𝑏𝑙) ↔ ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)))
54imbi2d 340 . . . . 5 (𝑥 = 𝐺 → ((𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙))))
63, 5raleqbidv 3317 . . . 4 (𝑥 = 𝐺 → (∀𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ ∀𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙))))
73, 6raleqbidv 3317 . . 3 (𝑥 = 𝐺 → (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ↔ ∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙))))
83rexeqdv 3312 . . . . 5 (𝑥 = 𝐺 → (∃𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
93, 8rexeqbidv 3318 . . . 4 (𝑥 = 𝐺 → (∃𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
109raleqbi1dv 3305 . . 3 (𝑥 = 𝐺 → (∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙)))
11 raleq 3307 . . . . . 6 (𝑥 = 𝐺 → (∀𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∀𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
123, 11rexeqbidv 3318 . . . . 5 (𝑥 = 𝐺 → (∃𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
133, 12rexeqbidv 3318 . . . 4 (𝑥 = 𝐺 → (∃𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
143, 13rexeqbidv 3318 . . 3 (𝑥 = 𝐺 → (∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙) ↔ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
157, 10, 143anbi123d 1436 . 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 29246 . 2 Plig = {𝑥 ∣ (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))}
1715, 16elab2g 3630 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wral 3062  wrex 3071  ∃!wreu 3349   cuni 4863  Pligcplig 29245
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-reu 3352  df-v 3445  df-in 3915  df-ss 3925  df-uni 4864  df-plig 29246
This theorem is referenced by:  ispligb  29248  tncp  29249  l2p  29250  eulplig  29256
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