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Theorem 3dim1lem5 39069
Description: Lemma for 3dim1 39070. (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 2993 . . 3 (𝑞 = 𝑢 → (𝑃𝑞𝑃𝑢))
2 oveq2 7427 . . . . 5 (𝑞 = 𝑢 → (𝑃 𝑞) = (𝑃 𝑢))
32breq2d 5161 . . . 4 (𝑞 = 𝑢 → (𝑟 (𝑃 𝑞) ↔ 𝑟 (𝑃 𝑢)))
43notbid 317 . . 3 (𝑞 = 𝑢 → (¬ 𝑟 (𝑃 𝑞) ↔ ¬ 𝑟 (𝑃 𝑢)))
52oveq1d 7434 . . . . 5 (𝑞 = 𝑢 → ((𝑃 𝑞) 𝑟) = ((𝑃 𝑢) 𝑟))
65breq2d 5161 . . . 4 (𝑞 = 𝑢 → (𝑠 ((𝑃 𝑞) 𝑟) ↔ 𝑠 ((𝑃 𝑢) 𝑟)))
76notbid 317 . . 3 (𝑞 = 𝑢 → (¬ 𝑠 ((𝑃 𝑞) 𝑟) ↔ ¬ 𝑠 ((𝑃 𝑢) 𝑟)))
81, 4, 73anbi123d 1432 . 2 (𝑞 = 𝑢 → ((𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)) ↔ (𝑃𝑢 ∧ ¬ 𝑟 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑟))))
9 breq1 5152 . . . 4 (𝑟 = 𝑣 → (𝑟 (𝑃 𝑢) ↔ 𝑣 (𝑃 𝑢)))
109notbid 317 . . 3 (𝑟 = 𝑣 → (¬ 𝑟 (𝑃 𝑢) ↔ ¬ 𝑣 (𝑃 𝑢)))
11 oveq2 7427 . . . . 5 (𝑟 = 𝑣 → ((𝑃 𝑢) 𝑟) = ((𝑃 𝑢) 𝑣))
1211breq2d 5161 . . . 4 (𝑟 = 𝑣 → (𝑠 ((𝑃 𝑢) 𝑟) ↔ 𝑠 ((𝑃 𝑢) 𝑣)))
1312notbid 317 . . 3 (𝑟 = 𝑣 → (¬ 𝑠 ((𝑃 𝑢) 𝑟) ↔ ¬ 𝑠 ((𝑃 𝑢) 𝑣)))
1410, 133anbi23d 1435 . 2 (𝑟 = 𝑣 → ((𝑃𝑢 ∧ ¬ 𝑟 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑟)) ↔ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑣))))
15 breq1 5152 . . . 4 (𝑠 = 𝑤 → (𝑠 ((𝑃 𝑢) 𝑣) ↔ 𝑤 ((𝑃 𝑢) 𝑣)))
1615notbid 317 . . 3 (𝑠 = 𝑤 → (¬ 𝑠 ((𝑃 𝑢) 𝑣) ↔ ¬ 𝑤 ((𝑃 𝑢) 𝑣)))
17163anbi3d 1438 . 2 (𝑠 = 𝑤 → ((𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑣)) ↔ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))))
188, 14, 17rspc3ev 3623 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 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wrex 3059   class class class wbr 5149  cfv 6549  (class class class)co 7419  lecple 17243  joincjn 18306  Atomscatm 38865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422
This theorem is referenced by:  3dim1  39070
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