| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | isdlat.b | . . . 4
⊢ 𝐵 = (Base‘𝐾) | 
| 2 |  | isdlat.j | . . . 4
⊢  ∨ =
(join‘𝐾) | 
| 3 |  | isdlat.m | . . . 4
⊢  ∧ =
(meet‘𝐾) | 
| 4 | 1, 2, 3 | isdlat 18568 | . . 3
⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))) | 
| 5 | 4 | simprbi 496 | . 2
⊢ (𝐾 ∈ DLat →
∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))) | 
| 6 |  | oveq1 7439 | . . . 4
⊢ (𝑥 = 𝑋 → (𝑥 ∧ (𝑦 ∨ 𝑧)) = (𝑋 ∧ (𝑦 ∨ 𝑧))) | 
| 7 |  | oveq1 7439 | . . . . 5
⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑦) = (𝑋 ∧ 𝑦)) | 
| 8 |  | oveq1 7439 | . . . . 5
⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑧) = (𝑋 ∧ 𝑧)) | 
| 9 | 7, 8 | oveq12d 7450 | . . . 4
⊢ (𝑥 = 𝑋 → ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧))) | 
| 10 | 6, 9 | eqeq12d 2752 | . . 3
⊢ (𝑥 = 𝑋 → ((𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑦 ∨ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)))) | 
| 11 |  | oveq1 7439 | . . . . 5
⊢ (𝑦 = 𝑌 → (𝑦 ∨ 𝑧) = (𝑌 ∨ 𝑧)) | 
| 12 | 11 | oveq2d 7448 | . . . 4
⊢ (𝑦 = 𝑌 → (𝑋 ∧ (𝑦 ∨ 𝑧)) = (𝑋 ∧ (𝑌 ∨ 𝑧))) | 
| 13 |  | oveq2 7440 | . . . . 5
⊢ (𝑦 = 𝑌 → (𝑋 ∧ 𝑦) = (𝑋 ∧ 𝑌)) | 
| 14 | 13 | oveq1d 7447 | . . . 4
⊢ (𝑦 = 𝑌 → ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧))) | 
| 15 | 12, 14 | eqeq12d 2752 | . . 3
⊢ (𝑦 = 𝑌 → ((𝑋 ∧ (𝑦 ∨ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑌 ∨ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)))) | 
| 16 |  | oveq2 7440 | . . . . 5
⊢ (𝑧 = 𝑍 → (𝑌 ∨ 𝑧) = (𝑌 ∨ 𝑍)) | 
| 17 | 16 | oveq2d 7448 | . . . 4
⊢ (𝑧 = 𝑍 → (𝑋 ∧ (𝑌 ∨ 𝑧)) = (𝑋 ∧ (𝑌 ∨ 𝑍))) | 
| 18 |  | oveq2 7440 | . . . . 5
⊢ (𝑧 = 𝑍 → (𝑋 ∧ 𝑧) = (𝑋 ∧ 𝑍)) | 
| 19 | 18 | oveq2d 7448 | . . . 4
⊢ (𝑧 = 𝑍 → ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍))) | 
| 20 | 17, 19 | eqeq12d 2752 | . . 3
⊢ (𝑧 = 𝑍 → ((𝑋 ∧ (𝑌 ∨ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)))) | 
| 21 | 10, 15, 20 | rspc3v 3637 | . 2
⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)))) | 
| 22 | 5, 21 | mpan9 506 | 1
⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍))) |