Step |
Hyp |
Ref |
Expression |
1 |
|
spsbc |
⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) → [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) ) ) |
2 |
|
sbcimg |
⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) ↔ ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ) ) |
3 |
|
eqid |
⊢ 𝐴 = 𝐴 |
4 |
|
eqsbc3 |
⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 ↔ 𝐴 = 𝐴 ) ) |
5 |
3 4
|
mpbiri |
⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 ) |
6 |
|
pm5.5 |
⊢ ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 → ( ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ↔ [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ) |
7 |
5 6
|
syl |
⊢ ( 𝐴 ∈ 𝑉 → ( ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ↔ [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ) |
8 |
|
eqsbc3 |
⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ↔ 𝐴 = 𝐵 ) ) |
9 |
2 7 8
|
3bitrd |
⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |
10 |
1 9
|
sylibd |
⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) ) |