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Theorem 3dim1lem5 38332
Description: Lemma for 3dim1 38333. (Contributed by NM, 26-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j ∨ = (joinβ€˜πΎ)
3dim0.l ≀ = (leβ€˜πΎ)
3dim0.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
3dim1lem5 (((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴) ∧ (𝑃 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑃 ∨ 𝑒) ∧ Β¬ 𝑀 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣))) β†’ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 (𝑃 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ π‘ž) ∧ Β¬ 𝑠 ≀ ((𝑃 ∨ π‘ž) ∨ π‘Ÿ)))
Distinct variable groups:   π‘Ÿ,π‘ž,𝑠,𝐴   ∨ ,π‘Ÿ,𝑠   𝑣,𝑒,𝑀,𝐴,π‘ž   ∨ ,π‘ž,𝑒,𝑣,𝑀   𝑒,𝐾,𝑣,𝑀   ≀ ,π‘ž   𝑒,π‘Ÿ,𝑣,𝑀, ≀ ,𝑠   𝑃,π‘ž,π‘Ÿ,𝑠,𝑒,𝑣,𝑀
Allowed substitution hints:   𝐾(𝑠,π‘Ÿ,π‘ž)

Proof of Theorem 3dim1lem5
StepHypRef Expression
1 neeq2 3004 . . 3 (π‘ž = 𝑒 β†’ (𝑃 β‰  π‘ž ↔ 𝑃 β‰  𝑒))
2 oveq2 7416 . . . . 5 (π‘ž = 𝑒 β†’ (𝑃 ∨ π‘ž) = (𝑃 ∨ 𝑒))
32breq2d 5160 . . . 4 (π‘ž = 𝑒 β†’ (π‘Ÿ ≀ (𝑃 ∨ π‘ž) ↔ π‘Ÿ ≀ (𝑃 ∨ 𝑒)))
43notbid 317 . . 3 (π‘ž = 𝑒 β†’ (Β¬ π‘Ÿ ≀ (𝑃 ∨ π‘ž) ↔ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑒)))
52oveq1d 7423 . . . . 5 (π‘ž = 𝑒 β†’ ((𝑃 ∨ π‘ž) ∨ π‘Ÿ) = ((𝑃 ∨ 𝑒) ∨ π‘Ÿ))
65breq2d 5160 . . . 4 (π‘ž = 𝑒 β†’ (𝑠 ≀ ((𝑃 ∨ π‘ž) ∨ π‘Ÿ) ↔ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ π‘Ÿ)))
76notbid 317 . . 3 (π‘ž = 𝑒 β†’ (Β¬ 𝑠 ≀ ((𝑃 ∨ π‘ž) ∨ π‘Ÿ) ↔ Β¬ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ π‘Ÿ)))
81, 4, 73anbi123d 1436 . 2 (π‘ž = 𝑒 β†’ ((𝑃 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ π‘ž) ∧ Β¬ 𝑠 ≀ ((𝑃 ∨ π‘ž) ∨ π‘Ÿ)) ↔ (𝑃 β‰  𝑒 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑒) ∧ Β¬ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ π‘Ÿ))))
9 breq1 5151 . . . 4 (π‘Ÿ = 𝑣 β†’ (π‘Ÿ ≀ (𝑃 ∨ 𝑒) ↔ 𝑣 ≀ (𝑃 ∨ 𝑒)))
109notbid 317 . . 3 (π‘Ÿ = 𝑣 β†’ (Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑒) ↔ Β¬ 𝑣 ≀ (𝑃 ∨ 𝑒)))
11 oveq2 7416 . . . . 5 (π‘Ÿ = 𝑣 β†’ ((𝑃 ∨ 𝑒) ∨ π‘Ÿ) = ((𝑃 ∨ 𝑒) ∨ 𝑣))
1211breq2d 5160 . . . 4 (π‘Ÿ = 𝑣 β†’ (𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ π‘Ÿ) ↔ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣)))
1312notbid 317 . . 3 (π‘Ÿ = 𝑣 β†’ (Β¬ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ π‘Ÿ) ↔ Β¬ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣)))
1410, 133anbi23d 1439 . 2 (π‘Ÿ = 𝑣 β†’ ((𝑃 β‰  𝑒 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑒) ∧ Β¬ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ π‘Ÿ)) ↔ (𝑃 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑃 ∨ 𝑒) ∧ Β¬ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣))))
15 breq1 5151 . . . 4 (𝑠 = 𝑀 β†’ (𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣) ↔ 𝑀 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣)))
1615notbid 317 . . 3 (𝑠 = 𝑀 β†’ (Β¬ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣) ↔ Β¬ 𝑀 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣)))
17163anbi3d 1442 . 2 (𝑠 = 𝑀 β†’ ((𝑃 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑃 ∨ 𝑒) ∧ Β¬ 𝑠 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣)) ↔ (𝑃 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑃 ∨ 𝑒) ∧ Β¬ 𝑀 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣))))
188, 14, 17rspc3ev 3628 1 (((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴) ∧ (𝑃 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑃 ∨ 𝑒) ∧ Β¬ 𝑀 ≀ ((𝑃 ∨ 𝑒) ∨ 𝑣))) β†’ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 (𝑃 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ π‘ž) ∧ Β¬ 𝑠 ≀ ((𝑃 ∨ π‘ž) ∨ π‘Ÿ)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  lecple 17203  joincjn 18263  Atomscatm 38128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411
This theorem is referenced by:  3dim1  38333
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