Step |
Hyp |
Ref |
Expression |
1 |
|
opelopabf.x |
⊢ Ⅎ 𝑥 𝜓 |
2 |
|
opelopabf.y |
⊢ Ⅎ 𝑦 𝜒 |
3 |
|
opelopabf.1 |
⊢ 𝐴 ∈ V |
4 |
|
opelopabf.2 |
⊢ 𝐵 ∈ V |
5 |
|
opelopabf.3 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
6 |
|
opelopabf.4 |
⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) ) |
7 |
|
opelopabsb |
⊢ ( 〈 𝐴 , 𝐵 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } ↔ [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ) |
8 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐵 |
9 |
8 1
|
nfsbcw |
⊢ Ⅎ 𝑥 [ 𝐵 / 𝑦 ] 𝜓 |
10 |
5
|
sbcbidv |
⊢ ( 𝑥 = 𝐴 → ( [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] 𝜓 ) ) |
11 |
9 10
|
sbciegf |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] 𝜓 ) ) |
12 |
3 11
|
ax-mp |
⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] 𝜓 ) |
13 |
2 6
|
sbciegf |
⊢ ( 𝐵 ∈ V → ( [ 𝐵 / 𝑦 ] 𝜓 ↔ 𝜒 ) ) |
14 |
4 13
|
ax-mp |
⊢ ( [ 𝐵 / 𝑦 ] 𝜓 ↔ 𝜒 ) |
15 |
7 12 14
|
3bitri |
⊢ ( 〈 𝐴 , 𝐵 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } ↔ 𝜒 ) |