Step |
Hyp |
Ref |
Expression |
1 |
|
csbeq1 |
⊢ ( 𝑦 = 𝐴 → ⦋ 𝑦 / 𝑥 ⦌ ( 𝐹 ''' 𝐵 ) = ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ''' 𝐵 ) ) |
2 |
|
csbeq1 |
⊢ ( 𝑦 = 𝐴 → ⦋ 𝑦 / 𝑥 ⦌ 𝐹 = ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ) |
3 |
|
csbeq1 |
⊢ ( 𝑦 = 𝐴 → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) |
4 |
2 3
|
afveq12d |
⊢ ( 𝑦 = 𝐴 → ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 ''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ''' ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) |
5 |
1 4
|
eqeq12d |
⊢ ( 𝑦 = 𝐴 → ( ⦋ 𝑦 / 𝑥 ⦌ ( 𝐹 ''' 𝐵 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 ''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ''' 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ''' ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) ) |
6 |
|
vex |
⊢ 𝑦 ∈ V |
7 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐹 |
8 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 |
9 |
7 8
|
nfafv |
⊢ Ⅎ 𝑥 ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 ''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
10 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑦 → 𝐹 = ⦋ 𝑦 / 𝑥 ⦌ 𝐹 ) |
11 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑦 → 𝐵 = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
12 |
10 11
|
afveq12d |
⊢ ( 𝑥 = 𝑦 → ( 𝐹 ''' 𝐵 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 ''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) ) |
13 |
6 9 12
|
csbief |
⊢ ⦋ 𝑦 / 𝑥 ⦌ ( 𝐹 ''' 𝐵 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 ''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
14 |
5 13
|
vtoclg |
⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ''' 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ''' ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) |