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Theorem cdleme29b 38867
Description: Transform cdleme28 38865. (Compare cdleme25b 38846.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b 𝐡 = (Baseβ€˜πΎ)
cdleme26.l ≀ = (leβ€˜πΎ)
cdleme26.j ∨ = (joinβ€˜πΎ)
cdleme26.m ∧ = (meetβ€˜πΎ)
cdleme26.a 𝐴 = (Atomsβ€˜πΎ)
cdleme26.h 𝐻 = (LHypβ€˜πΎ)
cdleme27.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme27.f 𝐹 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
cdleme27.z 𝑍 = ((𝑧 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ π‘Š)))
cdleme27.n 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)))
cdleme27.d 𝐷 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁))
cdleme27.c 𝐢 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
Assertion
Ref Expression
cdleme29b ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄 ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ βˆƒπ‘£ ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑣 = (𝐢 ∨ (𝑋 ∧ π‘Š))))
Distinct variable groups:   𝑒,𝑠,𝑧,𝐴   𝐡,𝑠,𝑒,𝑧   𝑒,𝐹   𝐻,𝑠,𝑧   ∨ ,𝑠,𝑒,𝑧   𝐾,𝑠,𝑧   ≀ ,𝑠,𝑒,𝑧   ∧ ,𝑠,𝑒,𝑧   𝑒,𝑁   𝑃,𝑠,𝑒,𝑧   𝑄,𝑠,𝑒,𝑧   π‘ˆ,𝑠,𝑒,𝑧   π‘Š,𝑠,𝑒,𝑧   𝑋,𝑠   𝑣,𝐴   𝑣,𝐡   𝑣, ∨   𝑣, ≀   𝑣, ∧   𝑣,𝑃   𝑣,𝑄   𝑣,π‘ˆ   𝑣,π‘Š   𝑣,𝐢   𝑣,𝑠,𝑍,𝑒   𝑧,𝑣,𝑋
Allowed substitution hints:   𝐢(𝑧,𝑒,𝑠)   𝐷(𝑧,𝑣,𝑒,𝑠)   𝐹(𝑧,𝑣,𝑠)   𝐻(𝑣,𝑒)   𝐾(𝑣,𝑒)   𝑁(𝑧,𝑣,𝑠)   𝑋(𝑒)   𝑍(𝑧)

Proof of Theorem cdleme29b
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 cdleme26.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 cdleme26.l . . 3 ≀ = (leβ€˜πΎ)
3 cdleme26.j . . 3 ∨ = (joinβ€˜πΎ)
4 cdleme26.m . . 3 ∧ = (meetβ€˜πΎ)
5 cdleme26.a . . 3 𝐴 = (Atomsβ€˜πΎ)
6 cdleme26.h . . 3 𝐻 = (LHypβ€˜πΎ)
7 cdleme27.u . . 3 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
8 cdleme27.f . . 3 𝐹 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
9 cdleme27.z . . 3 𝑍 = ((𝑧 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ π‘Š)))
10 cdleme27.n . . 3 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)))
11 cdleme27.d . . 3 𝐷 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁))
12 cdleme27.c . . 3 𝐢 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdleme29ex 38866 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄 ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ βˆƒπ‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (𝐢 ∨ (𝑋 ∧ π‘Š)) ∈ 𝐡))
14 eqid 2737 . . 3 ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))) = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
15 eqid 2737 . . 3 ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))) = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))
16 eqid 2737 . . 3 (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))) = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))))
17 eqid 2737 . . 3 if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) = if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))))
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17cdleme28 38865 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄 ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ βˆ€π‘  ∈ 𝐴 βˆ€π‘‘ ∈ 𝐴 (((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (Β¬ 𝑑 ≀ π‘Š ∧ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š))))
19 breq1 5113 . . . . . 6 (𝑠 = 𝑑 β†’ (𝑠 ≀ π‘Š ↔ 𝑑 ≀ π‘Š))
2019notbid 318 . . . . 5 (𝑠 = 𝑑 β†’ (Β¬ 𝑠 ≀ π‘Š ↔ Β¬ 𝑑 ≀ π‘Š))
21 oveq1 7369 . . . . . 6 (𝑠 = 𝑑 β†’ (𝑠 ∨ (𝑋 ∧ π‘Š)) = (𝑑 ∨ (𝑋 ∧ π‘Š)))
2221eqeq1d 2739 . . . . 5 (𝑠 = 𝑑 β†’ ((𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋 ↔ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋))
2320, 22anbi12d 632 . . . 4 (𝑠 = 𝑑 β†’ ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ↔ (Β¬ 𝑑 ≀ π‘Š ∧ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋)))
2412oveq1i 7372 . . . . 5 (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹) ∨ (𝑋 ∧ π‘Š))
25 breq1 5113 . . . . . . 7 (𝑠 = 𝑑 β†’ (𝑠 ≀ (𝑃 ∨ 𝑄) ↔ 𝑑 ≀ (𝑃 ∨ 𝑄)))
26 oveq1 7369 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑑 β†’ (𝑠 ∨ 𝑧) = (𝑑 ∨ 𝑧))
2726oveq1d 7377 . . . . . . . . . . . . . . 15 (𝑠 = 𝑑 β†’ ((𝑠 ∨ 𝑧) ∧ π‘Š) = ((𝑑 ∨ 𝑧) ∧ π‘Š))
2827oveq2d 7378 . . . . . . . . . . . . . 14 (𝑠 = 𝑑 β†’ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)) = (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))
2928oveq2d 7378 . . . . . . . . . . . . 13 (𝑠 = 𝑑 β†’ ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š))) = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))
3010, 29eqtrid 2789 . . . . . . . . . . . 12 (𝑠 = 𝑑 β†’ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))
3130eqeq2d 2748 . . . . . . . . . . 11 (𝑠 = 𝑑 β†’ (𝑒 = 𝑁 ↔ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))))
3231imbi2d 341 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ (((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁) ↔ ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))))
3332ralbidv 3175 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁) ↔ βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))))
3433riotabidv 7320 . . . . . . . 8 (𝑠 = 𝑑 β†’ (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁)) = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))))
3511, 34eqtrid 2789 . . . . . . 7 (𝑠 = 𝑑 β†’ 𝐷 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))))
36 oveq1 7369 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (𝑠 ∨ π‘ˆ) = (𝑑 ∨ π‘ˆ))
37 oveq2 7370 . . . . . . . . . . 11 (𝑠 = 𝑑 β†’ (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑑))
3837oveq1d 7377 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ ((𝑃 ∨ 𝑠) ∧ π‘Š) = ((𝑃 ∨ 𝑑) ∧ π‘Š))
3938oveq2d 7378 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)) = (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
4036, 39oveq12d 7380 . . . . . . . 8 (𝑠 = 𝑑 β†’ ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š))) = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))))
418, 40eqtrid 2789 . . . . . . 7 (𝑠 = 𝑑 β†’ 𝐹 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))))
4225, 35, 41ifbieq12d 4519 . . . . . 6 (𝑠 = 𝑑 β†’ if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹) = if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))))
4342oveq1d 7377 . . . . 5 (𝑠 = 𝑑 β†’ (if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹) ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š)))
4424, 43eqtrid 2789 . . . 4 (𝑠 = 𝑑 β†’ (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š)))
4523, 44reusv3 5365 . . 3 (βˆƒπ‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (𝐢 ∨ (𝑋 ∧ π‘Š)) ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝐴 βˆ€π‘‘ ∈ 𝐴 (((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (Β¬ 𝑑 ≀ π‘Š ∧ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š))) ↔ βˆƒπ‘£ ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑣 = (𝐢 ∨ (𝑋 ∧ π‘Š)))))
4645biimpd 228 . 2 (βˆƒπ‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (𝐢 ∨ (𝑋 ∧ π‘Š)) ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝐴 βˆ€π‘‘ ∈ 𝐴 (((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (Β¬ 𝑑 ≀ π‘Š ∧ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š))) β†’ βˆƒπ‘£ ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑣 = (𝐢 ∨ (𝑋 ∧ π‘Š)))))
4713, 18, 46sylc 65 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄 ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ βˆƒπ‘£ ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑣 = (𝐢 ∨ (𝑋 ∧ π‘Š))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  ifcif 4491   class class class wbr 5110  β€˜cfv 6501  β„©crio 7317  (class class class)co 7362  Basecbs 17090  lecple 17147  joincjn 18207  meetcmee 18208  Atomscatm 37754  HLchlt 37841  LHypclh 38476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-riotaBAD 37444
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-undef 8209  df-proset 18191  df-poset 18209  df-plt 18226  df-lub 18242  df-glb 18243  df-join 18244  df-meet 18245  df-p0 18321  df-p1 18322  df-lat 18328  df-clat 18395  df-oposet 37667  df-ol 37669  df-oml 37670  df-covers 37757  df-ats 37758  df-atl 37789  df-cvlat 37813  df-hlat 37842  df-llines 37990  df-lplanes 37991  df-lvols 37992  df-lines 37993  df-psubsp 37995  df-pmap 37996  df-padd 38288  df-lhyp 38480
This theorem is referenced by:  cdleme29c  38868
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