Step |
Hyp |
Ref |
Expression |
1 |
|
doch11.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
doch11.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
doch11.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
doch11.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
5 |
|
doch11.x |
⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 ) |
6 |
|
doch11.y |
⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 ) |
7 |
1 2 3 4 6 5
|
dochord |
⊢ ( 𝜑 → ( 𝑌 ⊆ 𝑋 ↔ ( ⊥ ‘ 𝑋 ) ⊆ ( ⊥ ‘ 𝑌 ) ) ) |
8 |
1 2 3 4 5 6
|
dochord |
⊢ ( 𝜑 → ( 𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) ) |
9 |
7 8
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑌 ) ↔ ( ( ⊥ ‘ 𝑋 ) ⊆ ( ⊥ ‘ 𝑌 ) ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) ) ) |
10 |
|
eqcom |
⊢ ( 𝑋 = 𝑌 ↔ 𝑌 = 𝑋 ) |
11 |
|
eqss |
⊢ ( 𝑌 = 𝑋 ↔ ( 𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑌 ) ) |
12 |
10 11
|
bitri |
⊢ ( 𝑋 = 𝑌 ↔ ( 𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑌 ) ) |
13 |
|
eqss |
⊢ ( ( ⊥ ‘ 𝑋 ) = ( ⊥ ‘ 𝑌 ) ↔ ( ( ⊥ ‘ 𝑋 ) ⊆ ( ⊥ ‘ 𝑌 ) ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) ) |
14 |
9 12 13
|
3bitr4g |
⊢ ( 𝜑 → ( 𝑋 = 𝑌 ↔ ( ⊥ ‘ 𝑋 ) = ( ⊥ ‘ 𝑌 ) ) ) |
15 |
14
|
bicomd |
⊢ ( 𝜑 → ( ( ⊥ ‘ 𝑋 ) = ( ⊥ ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) |