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Theorem eulplig 27569
Description: Through two distinct points of a planar incidence geometry, there is a unique line. (Contributed by BJ, 2-Dec-2021.)
Hypothesis
Ref Expression
eulplig.1 𝑃 = 𝐺
Assertion
Ref Expression
eulplig ((𝐺 ∈ Plig ∧ ((𝐴𝑃𝐵𝑃) ∧ 𝐴𝐵)) → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙))
Distinct variable groups:   𝐺,𝑙   𝐴,𝑙   𝐵,𝑙
Allowed substitution hint:   𝑃(𝑙)

Proof of Theorem eulplig
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eulplig.1 . . . . 5 𝑃 = 𝐺
21isplig 27560 . . . 4 (𝐺 ∈ Plig → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
32ibi 256 . . 3 (𝐺 ∈ Plig → (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
4 simp1 1128 . . 3 ((∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)) → ∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)))
5 simpl 474 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
6 simpr 479 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑏 = 𝐵)
75, 6neeq12d 2957 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝑏𝐴𝐵))
8 eleq1 2791 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎𝑙𝐴𝑙))
9 eleq1 2791 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏𝑙𝐵𝑙))
108, 9bi2anan9 953 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝑙𝑏𝑙) ↔ (𝐴𝑙𝐵𝑙)))
1110reubidv 3229 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐵) → (∃!𝑙𝐺 (𝑎𝑙𝑏𝑙) ↔ ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙)))
127, 11imbi12d 333 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ↔ (𝐴𝐵 → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙))))
1312rspc2gv 3425 . . . . . 6 ((𝐴𝑃𝐵𝑃) → (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) → (𝐴𝐵 → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙))))
1413com23 86 . . . . 5 ((𝐴𝑃𝐵𝑃) → (𝐴𝐵 → (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙))))
1514imp 444 . . . 4 (((𝐴𝑃𝐵𝑃) ∧ 𝐴𝐵) → (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙)))
1615com12 32 . . 3 (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) → (((𝐴𝑃𝐵𝑃) ∧ 𝐴𝐵) → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙)))
173, 4, 163syl 18 . 2 (𝐺 ∈ Plig → (((𝐴𝑃𝐵𝑃) ∧ 𝐴𝐵) → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙)))
1817imp 444 1 ((𝐺 ∈ Plig ∧ ((𝐴𝑃𝐵𝑃) ∧ 𝐴𝐵)) → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1596  wcel 2103  wne 2896  wral 3014  wrex 3015  ∃!wreu 3016   cuni 4544  Pligcplig 27558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-v 3306  df-uni 4545  df-plig 27559
This theorem is referenced by: (None)
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