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Theorem dalem62 37404
Description: Lemma for dath 37406. Eliminate the condition 𝜓 containing dummy variables 𝑐 and 𝑑. (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalem62.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalem62.l = (le‘𝐾)
dalem62.j = (join‘𝐾)
dalem62.a 𝐴 = (Atoms‘𝐾)
dalem62.m = (meet‘𝐾)
dalem62.o 𝑂 = (LPlanes‘𝐾)
dalem62.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem62.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem62.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dalem62.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
dalem62.f 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
Assertion
Ref Expression
dalem62 ((𝜑𝑌 = 𝑍) → 𝐹 (𝐷 𝐸))

Proof of Theorem dalem62
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dalem62.ph . . 3 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
2 dalem62.l . . 3 = (le‘𝐾)
3 dalem62.j . . 3 = (join‘𝐾)
4 dalem62.a . . 3 𝐴 = (Atoms‘𝐾)
5 biid 264 . . 3 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
6 dalem62.o . . 3 𝑂 = (LPlanes‘𝐾)
7 dalem62.y . . 3 𝑌 = ((𝑃 𝑄) 𝑅)
8 dalem62.z . . 3 𝑍 = ((𝑆 𝑇) 𝑈)
91, 2, 3, 4, 5, 6, 7, 8dalem20 37363 . 2 ((𝜑𝑌 = 𝑍) → ∃𝑐𝑑((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
10 dalem62.m . . . . 5 = (meet‘𝐾)
11 dalem62.d . . . . 5 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
12 dalem62.e . . . . 5 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
13 dalem62.f . . . . 5 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
141, 2, 3, 4, 5, 10, 6, 7, 8, 11, 12, 13dalem61 37403 . . . 4 ((𝜑𝑌 = 𝑍 ∧ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑)))) → 𝐹 (𝐷 𝐸))
15143expia 1122 . . 3 ((𝜑𝑌 = 𝑍) → (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝐹 (𝐷 𝐸)))
1615exlimdvv 1941 . 2 ((𝜑𝑌 = 𝑍) → (∃𝑐𝑑((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝐹 (𝐷 𝐸)))
179, 16mpd 15 1 ((𝜑𝑌 = 𝑍) → 𝐹 (𝐷 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  wne 2935   class class class wbr 5040  cfv 6350  (class class class)co 7183  Basecbs 16599  lecple 16688  joincjn 17683  meetcmee 17684  Atomscatm 36933  HLchlt 37020  LPlanesclpl 37162
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 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7140  df-ov 7186  df-oprab 7187  df-proset 17667  df-poset 17685  df-plt 17697  df-lub 17713  df-glb 17714  df-join 17715  df-meet 17716  df-p0 17778  df-p1 17779  df-lat 17785  df-clat 17847  df-oposet 36846  df-ol 36848  df-oml 36849  df-covers 36936  df-ats 36937  df-atl 36968  df-cvlat 36992  df-hlat 37021  df-llines 37168  df-lplanes 37169  df-lvols 37170
This theorem is referenced by:  dalem63  37405
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