Step |
Hyp |
Ref |
Expression |
1 |
|
caovcan.1 |
⊢ 𝐶 ∈ V |
2 |
|
caovcan.2 |
⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ) |
3 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑦 ) ) |
4 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝐶 ) = ( 𝐴 𝐹 𝐶 ) ) |
5 |
3 4
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) ↔ ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) ) ) |
6 |
5
|
imbi1d |
⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ↔ ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) ) |
7 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) ) |
8 |
7
|
eqeq1d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) ↔ ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ) ) |
9 |
|
eqeq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 = 𝐶 ↔ 𝐵 = 𝐶 ) ) |
10 |
8 9
|
imbi12d |
⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ↔ ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) ) ) |
11 |
|
oveq2 |
⊢ ( 𝑧 = 𝐶 → ( 𝑥 𝐹 𝑧 ) = ( 𝑥 𝐹 𝐶 ) ) |
12 |
11
|
eqeq2d |
⊢ ( 𝑧 = 𝐶 → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) ) ) |
13 |
|
eqeq2 |
⊢ ( 𝑧 = 𝐶 → ( 𝑦 = 𝑧 ↔ 𝑦 = 𝐶 ) ) |
14 |
12 13
|
imbi12d |
⊢ ( 𝑧 = 𝐶 → ( ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ↔ ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) ) |
15 |
14
|
imbi2d |
⊢ ( 𝑧 = 𝐶 → ( ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ) ↔ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) ) ) |
16 |
1 15 2
|
vtocl |
⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) |
17 |
6 10 16
|
vtocl2ga |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) ) |