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Theorem 4atex 39451
Description: Whenever there are at least 4 atoms under 𝑃 ∨ 𝑄 (specifically, 𝑃, 𝑄, π‘Ÿ, and (𝑃 ∨ 𝑄) ∧ π‘Š), there are also at least 4 atoms under 𝑃 ∨ 𝑆. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p ∨ q/0 and hence p ∨ s/0 contains at least four atoms..." Note that by cvlsupr2 38717, our (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ) is a shorter way to express π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄). (Contributed by NM, 27-May-2013.)
Hypotheses
Ref Expression
4that.l ≀ = (leβ€˜πΎ)
4that.j ∨ = (joinβ€˜πΎ)
4that.a 𝐴 = (Atomsβ€˜πΎ)
4that.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
4atex (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
Distinct variable groups:   𝑧,π‘Ÿ,𝐴   𝐻,π‘Ÿ   ∨ ,π‘Ÿ,𝑧   𝐾,π‘Ÿ,𝑧   ≀ ,π‘Ÿ,𝑧   𝑃,π‘Ÿ,𝑧   𝑄,π‘Ÿ,𝑧   𝑆,π‘Ÿ,𝑧   π‘Š,π‘Ÿ,𝑧
Allowed substitution hint:   𝐻(𝑧)

Proof of Theorem 4atex
StepHypRef Expression
1 simp21l 1287 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑃 ∈ 𝐴)
21ad2antrr 723 . . . 4 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 = 𝑃) β†’ 𝑃 ∈ 𝐴)
3 simp21r 1288 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ Β¬ 𝑃 ≀ π‘Š)
43ad2antrr 723 . . . 4 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 = 𝑃) β†’ Β¬ 𝑃 ≀ π‘Š)
5 oveq1 7409 . . . . . 6 (𝑃 = 𝑆 β†’ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))
65eqcoms 2732 . . . . 5 (𝑆 = 𝑃 β†’ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))
76adantl 481 . . . 4 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 = 𝑃) β†’ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))
8 breq1 5142 . . . . . . 7 (𝑧 = 𝑃 β†’ (𝑧 ≀ π‘Š ↔ 𝑃 ≀ π‘Š))
98notbid 318 . . . . . 6 (𝑧 = 𝑃 β†’ (Β¬ 𝑧 ≀ π‘Š ↔ Β¬ 𝑃 ≀ π‘Š))
10 oveq2 7410 . . . . . . 7 (𝑧 = 𝑃 β†’ (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝑃))
11 oveq2 7410 . . . . . . 7 (𝑧 = 𝑃 β†’ (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑃))
1210, 11eqeq12d 2740 . . . . . 6 (𝑧 = 𝑃 β†’ ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃)))
139, 12anbi12d 630 . . . . 5 (𝑧 = 𝑃 β†’ ((Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (Β¬ 𝑃 ≀ π‘Š ∧ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))))
1413rspcev 3604 . . . 4 ((𝑃 ∈ 𝐴 ∧ (Β¬ 𝑃 ≀ π‘Š ∧ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
152, 4, 7, 14syl12anc 834 . . 3 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 = 𝑃) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
16 simpl3r 1226 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))
1716ad2antrr 723 . . . . 5 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 = 𝑄) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))
18 breq1 5142 . . . . . . . . . 10 (π‘Ÿ = 𝑧 β†’ (π‘Ÿ ≀ π‘Š ↔ 𝑧 ≀ π‘Š))
1918notbid 318 . . . . . . . . 9 (π‘Ÿ = 𝑧 β†’ (Β¬ π‘Ÿ ≀ π‘Š ↔ Β¬ 𝑧 ≀ π‘Š))
20 oveq2 7410 . . . . . . . . . 10 (π‘Ÿ = 𝑧 β†’ (𝑃 ∨ π‘Ÿ) = (𝑃 ∨ 𝑧))
21 oveq2 7410 . . . . . . . . . 10 (π‘Ÿ = 𝑧 β†’ (𝑄 ∨ π‘Ÿ) = (𝑄 ∨ 𝑧))
2220, 21eqeq12d 2740 . . . . . . . . 9 (π‘Ÿ = 𝑧 β†’ ((𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ) ↔ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
2319, 22anbi12d 630 . . . . . . . 8 (π‘Ÿ = 𝑧 β†’ ((Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ↔ (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
2423cbvrexvw 3227 . . . . . . 7 (βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ↔ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
25 oveq1 7409 . . . . . . . . . 10 (𝑆 = 𝑄 β†’ (𝑆 ∨ 𝑧) = (𝑄 ∨ 𝑧))
2625eqeq2d 2735 . . . . . . . . 9 (𝑆 = 𝑄 β†’ ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
2726anbi2d 628 . . . . . . . 8 (𝑆 = 𝑄 β†’ ((Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
2827rexbidv 3170 . . . . . . 7 (𝑆 = 𝑄 β†’ (βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
2924, 28bitr4id 290 . . . . . 6 (𝑆 = 𝑄 β†’ (βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ↔ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))))
3029adantl 481 . . . . 5 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 = 𝑄) β†’ (βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ↔ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))))
3117, 30mpbid 231 . . . 4 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 = 𝑄) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
32 simp22l 1289 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑄 ∈ 𝐴)
3332ad3antrrr 727 . . . . 5 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑄 ∈ 𝐴)
34 simp22r 1290 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ Β¬ 𝑄 ≀ π‘Š)
3534ad3antrrr 727 . . . . 5 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ Β¬ 𝑄 ≀ π‘Š)
36 simp3l 1198 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑃 β‰  𝑄)
3736necomd 2988 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑄 β‰  𝑃)
3837ad3antrrr 727 . . . . . 6 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑄 β‰  𝑃)
39 simpr 484 . . . . . . 7 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑆 β‰  𝑄)
4039necomd 2988 . . . . . 6 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑄 β‰  𝑆)
41 simpllr 773 . . . . . . 7 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄))
42 simp1l 1194 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ HL)
43 hlcvl 38733 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ CvLat)
4442, 43syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ CvLat)
4544ad3antrrr 727 . . . . . . . 8 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝐾 ∈ CvLat)
46 simp23 1205 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ 𝑆 ∈ 𝐴)
4746ad3antrrr 727 . . . . . . . 8 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑆 ∈ 𝐴)
481ad3antrrr 727 . . . . . . . 8 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑃 ∈ 𝐴)
49 simplr 766 . . . . . . . 8 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑆 β‰  𝑃)
50 4that.l . . . . . . . . 9 ≀ = (leβ€˜πΎ)
51 4that.j . . . . . . . . 9 ∨ = (joinβ€˜πΎ)
52 4that.a . . . . . . . . 9 𝐴 = (Atomsβ€˜πΎ)
5350, 51, 52cvlatexch1 38710 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑆 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑆 β‰  𝑃) β†’ (𝑆 ≀ (𝑃 ∨ 𝑄) β†’ 𝑄 ≀ (𝑃 ∨ 𝑆)))
5445, 47, 33, 48, 49, 53syl131anc 1380 . . . . . . 7 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ (𝑆 ≀ (𝑃 ∨ 𝑄) β†’ 𝑄 ≀ (𝑃 ∨ 𝑆)))
5541, 54mpd 15 . . . . . 6 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑄 ≀ (𝑃 ∨ 𝑆))
5649necomd 2988 . . . . . . 7 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ 𝑃 β‰  𝑆)
5752, 50, 51cvlsupr2 38717 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑆) β†’ ((𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄) ↔ (𝑄 β‰  𝑃 ∧ 𝑄 β‰  𝑆 ∧ 𝑄 ≀ (𝑃 ∨ 𝑆))))
5845, 48, 47, 33, 56, 57syl131anc 1380 . . . . . 6 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ ((𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄) ↔ (𝑄 β‰  𝑃 ∧ 𝑄 β‰  𝑆 ∧ 𝑄 ≀ (𝑃 ∨ 𝑆))))
5938, 40, 55, 58mpbir3and 1339 . . . . 5 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄))
60 breq1 5142 . . . . . . . 8 (𝑧 = 𝑄 β†’ (𝑧 ≀ π‘Š ↔ 𝑄 ≀ π‘Š))
6160notbid 318 . . . . . . 7 (𝑧 = 𝑄 β†’ (Β¬ 𝑧 ≀ π‘Š ↔ Β¬ 𝑄 ≀ π‘Š))
62 oveq2 7410 . . . . . . . 8 (𝑧 = 𝑄 β†’ (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝑄))
63 oveq2 7410 . . . . . . . 8 (𝑧 = 𝑄 β†’ (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑄))
6462, 63eqeq12d 2740 . . . . . . 7 (𝑧 = 𝑄 β†’ ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄)))
6561, 64anbi12d 630 . . . . . 6 (𝑧 = 𝑄 β†’ ((Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (Β¬ 𝑄 ≀ π‘Š ∧ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄))))
6665rspcev 3604 . . . . 5 ((𝑄 ∈ 𝐴 ∧ (Β¬ 𝑄 ≀ π‘Š ∧ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
6733, 35, 59, 66syl12anc 834 . . . 4 ((((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) ∧ 𝑆 β‰  𝑄) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
6831, 67pm2.61dane 3021 . . 3 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ 𝑆 β‰  𝑃) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
6915, 68pm2.61dane 3021 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
70 simpl1 1188 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
71 simpl2 1189 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴))
72 simpl3l 1225 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑃 β‰  𝑄)
73 simpr 484 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))
74 simpl3r 1226 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))
75 4that.h . . . 4 𝐻 = (LHypβ€˜πΎ)
7650, 51, 52, 754atexlem7 39450 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
7770, 71, 72, 73, 74, 76syl113anc 1379 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄)) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
7869, 77pm2.61dan 810 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆƒwrex 3062   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402  lecple 17209  joincjn 18272  Atomscatm 38637  CvLatclc 38639  HLchlt 38724  LHypclh 39359
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-proset 18256  df-poset 18274  df-plt 18291  df-lub 18307  df-glb 18308  df-join 18309  df-meet 18310  df-p0 18386  df-p1 18387  df-lat 18393  df-clat 18460  df-oposet 38550  df-ol 38552  df-oml 38553  df-covers 38640  df-ats 38641  df-atl 38672  df-cvlat 38696  df-hlat 38725  df-llines 38873  df-lplanes 38874  df-lhyp 39363
This theorem is referenced by:  4atex2  39452
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