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Theorem prslem 17650
Description: Lemma for prsref 17651 and prstr 17652. (Contributed by Mario Carneiro, 1-Feb-2015.)
Hypotheses
Ref Expression
isprs.b 𝐵 = (Base‘𝐾)
isprs.l = (le‘𝐾)
Assertion
Ref Expression
prslem ((𝐾 ∈ Proset ∧ (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 17649 . . 3 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
43simprbi 500 . 2 (𝐾 ∈ Proset → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
5 breq12 5032 . . . . 5 ((𝑥 = 𝑋𝑥 = 𝑋) → (𝑥 𝑥𝑋 𝑋))
65anidms 570 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑥𝑋 𝑋))
7 breq1 5030 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
87anbi1d 633 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑋 𝑦𝑦 𝑧)))
9 breq1 5030 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑧𝑋 𝑧))
108, 9imbi12d 348 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)))
116, 10anbi12d 634 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧))))
12 breq2 5031 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
13 breq1 5030 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
1412, 13anbi12d 634 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑧) ↔ (𝑋 𝑌𝑌 𝑧)))
1514imbi1d 345 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)))
1615anbi2d 632 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧))))
17 breq2 5031 . . . . . 6 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
1817anbi2d 632 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌𝑌 𝑧) ↔ (𝑋 𝑌𝑌 𝑍)))
19 breq2 5031 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
2018, 19imbi12d 348 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
2120anbi2d 632 . . 3 (𝑧 = 𝑍 → ((𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
2211, 16, 21rspc3v 3537 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
234, 22mpan9 510 1 ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wral 3053  Vcvv 3397   class class class wbr 5027  cfv 6333  Basecbs 16579  lecple 16668   Proset cproset 17645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-nul 5171
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-iota 6291  df-fv 6341  df-proset 17647
This theorem is referenced by:  prsref  17651  prstr  17652
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