Step |
Hyp |
Ref |
Expression |
1 |
|
caovordg.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ) |
2 |
1
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ) |
3 |
|
breq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝑅 𝑦 ↔ 𝐴 𝑅 𝑦 ) ) |
4 |
|
oveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑧 𝐹 𝑥 ) = ( 𝑧 𝐹 𝐴 ) ) |
5 |
4
|
breq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ) |
6 |
3 5
|
bibi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ↔ ( 𝐴 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ) ) |
7 |
|
breq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝑅 𝑦 ↔ 𝐴 𝑅 𝐵 ) ) |
8 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝑧 𝐹 𝑦 ) = ( 𝑧 𝐹 𝐵 ) ) |
9 |
8
|
breq2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝐵 ) ) ) |
10 |
7 9
|
bibi12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ↔ ( 𝐴 𝑅 𝐵 ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝐵 ) ) ) ) |
11 |
|
oveq1 |
⊢ ( 𝑧 = 𝐶 → ( 𝑧 𝐹 𝐴 ) = ( 𝐶 𝐹 𝐴 ) ) |
12 |
|
oveq1 |
⊢ ( 𝑧 = 𝐶 → ( 𝑧 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐵 ) ) |
13 |
11 12
|
breq12d |
⊢ ( 𝑧 = 𝐶 → ( ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝐵 ) ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) ) |
14 |
13
|
bibi2d |
⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝑅 𝐵 ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝐵 ) ) ↔ ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) ) ) |
15 |
6 10 14
|
rspc3v |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) ) ) |
16 |
2 15
|
mpan9 |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) ) |