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Theorem 3dim1lem5 39468
Description: Lemma for 3dim1 39469. (Contributed by NM, 26-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j = (join‘𝐾)
3dim0.l = (le‘𝐾)
3dim0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
3dim1lem5 (((𝑢𝐴𝑣𝐴𝑤𝐴) ∧ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)))
Distinct variable groups:   𝑟,𝑞,𝑠,𝐴   ,𝑟,𝑠   𝑣,𝑢,𝑤,𝐴,𝑞   ,𝑞,𝑢,𝑣,𝑤   𝑢,𝐾,𝑣,𝑤   ,𝑞   𝑢,𝑟,𝑣,𝑤, ,𝑠   𝑃,𝑞,𝑟,𝑠,𝑢,𝑣,𝑤
Allowed substitution hints:   𝐾(𝑠,𝑟,𝑞)

Proof of Theorem 3dim1lem5
StepHypRef Expression
1 neeq2 3004 . . 3 (𝑞 = 𝑢 → (𝑃𝑞𝑃𝑢))
2 oveq2 7439 . . . . 5 (𝑞 = 𝑢 → (𝑃 𝑞) = (𝑃 𝑢))
32breq2d 5155 . . . 4 (𝑞 = 𝑢 → (𝑟 (𝑃 𝑞) ↔ 𝑟 (𝑃 𝑢)))
43notbid 318 . . 3 (𝑞 = 𝑢 → (¬ 𝑟 (𝑃 𝑞) ↔ ¬ 𝑟 (𝑃 𝑢)))
52oveq1d 7446 . . . . 5 (𝑞 = 𝑢 → ((𝑃 𝑞) 𝑟) = ((𝑃 𝑢) 𝑟))
65breq2d 5155 . . . 4 (𝑞 = 𝑢 → (𝑠 ((𝑃 𝑞) 𝑟) ↔ 𝑠 ((𝑃 𝑢) 𝑟)))
76notbid 318 . . 3 (𝑞 = 𝑢 → (¬ 𝑠 ((𝑃 𝑞) 𝑟) ↔ ¬ 𝑠 ((𝑃 𝑢) 𝑟)))
81, 4, 73anbi123d 1438 . 2 (𝑞 = 𝑢 → ((𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)) ↔ (𝑃𝑢 ∧ ¬ 𝑟 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑟))))
9 breq1 5146 . . . 4 (𝑟 = 𝑣 → (𝑟 (𝑃 𝑢) ↔ 𝑣 (𝑃 𝑢)))
109notbid 318 . . 3 (𝑟 = 𝑣 → (¬ 𝑟 (𝑃 𝑢) ↔ ¬ 𝑣 (𝑃 𝑢)))
11 oveq2 7439 . . . . 5 (𝑟 = 𝑣 → ((𝑃 𝑢) 𝑟) = ((𝑃 𝑢) 𝑣))
1211breq2d 5155 . . . 4 (𝑟 = 𝑣 → (𝑠 ((𝑃 𝑢) 𝑟) ↔ 𝑠 ((𝑃 𝑢) 𝑣)))
1312notbid 318 . . 3 (𝑟 = 𝑣 → (¬ 𝑠 ((𝑃 𝑢) 𝑟) ↔ ¬ 𝑠 ((𝑃 𝑢) 𝑣)))
1410, 133anbi23d 1441 . 2 (𝑟 = 𝑣 → ((𝑃𝑢 ∧ ¬ 𝑟 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑟)) ↔ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑣))))
15 breq1 5146 . . . 4 (𝑠 = 𝑤 → (𝑠 ((𝑃 𝑢) 𝑣) ↔ 𝑤 ((𝑃 𝑢) 𝑣)))
1615notbid 318 . . 3 (𝑠 = 𝑤 → (¬ 𝑠 ((𝑃 𝑢) 𝑣) ↔ ¬ 𝑤 ((𝑃 𝑢) 𝑣)))
17163anbi3d 1444 . 2 (𝑠 = 𝑤 → ((𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑣)) ↔ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))))
188, 14, 17rspc3ev 3639 1 (((𝑢𝐴𝑣𝐴𝑤𝐴) ∧ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  Atomscatm 39264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  3dim1  39469
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