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Theorem prslem 16754
 Description: Lemma for prsref 16755 and prstr 16756. (Contributed by Mario Carneiro, 1-Feb-2015.)
Hypotheses
Ref Expression
isprs.b 𝐵 = (Base‘𝐾)
isprs.l = (le‘𝐾)
Assertion
Ref Expression
prslem ((𝐾 ∈ Preset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))

Proof of Theorem prslem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isprs.b . . . 4 𝐵 = (Base‘𝐾)
2 isprs.l . . . 4 = (le‘𝐾)
31, 2isprs 16753 . . 3 (𝐾 ∈ Preset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
43simprbi 479 . 2 (𝐾 ∈ Preset → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
5 breq12 4588 . . . . 5 ((𝑥 = 𝑋𝑥 = 𝑋) → (𝑥 𝑥𝑋 𝑋))
65anidms 675 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑥𝑋 𝑋))
7 breq1 4586 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
87anbi1d 737 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑋 𝑦𝑦 𝑧)))
9 breq1 4586 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑧𝑋 𝑧))
108, 9imbi12d 333 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)))
116, 10anbi12d 743 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧))))
12 breq2 4587 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
13 breq1 4586 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
1412, 13anbi12d 743 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑧) ↔ (𝑋 𝑌𝑌 𝑧)))
1514imbi1d 330 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)))
1615anbi2d 736 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧))))
17 breq2 4587 . . . . . 6 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
1817anbi2d 736 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌𝑌 𝑧) ↔ (𝑋 𝑌𝑌 𝑍)))
19 breq2 4587 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
2018, 19imbi12d 333 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
2120anbi2d 736 . . 3 (𝑧 = 𝑍 → ((𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
2211, 16, 21rspc3v 3296 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
234, 22mpan9 485 1 ((𝐾 ∈ Preset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775   Preset cpreset 16749 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  ax-nul 4717 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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-preset 16751 This theorem is referenced by:  prsref  16755  prstr  16756
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