Step |
Hyp |
Ref |
Expression |
1 |
|
nfsbc1v |
⊢ Ⅎ 𝑦 [ ( 1st ‘ ( 1st ‘ 𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) / 𝑧 ] [ ( 2nd ‘ 𝑥 ) / 𝑤 ] 𝜑 |
2 |
|
nfcv |
⊢ Ⅎ 𝑧 ( 1st ‘ ( 1st ‘ 𝑥 ) ) |
3 |
|
nfsbc1v |
⊢ Ⅎ 𝑧 [ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) / 𝑧 ] [ ( 2nd ‘ 𝑥 ) / 𝑤 ] 𝜑 |
4 |
2 3
|
nfsbcw |
⊢ Ⅎ 𝑧 [ ( 1st ‘ ( 1st ‘ 𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) / 𝑧 ] [ ( 2nd ‘ 𝑥 ) / 𝑤 ] 𝜑 |
5 |
|
nfcv |
⊢ Ⅎ 𝑤 ( 1st ‘ ( 1st ‘ 𝑥 ) ) |
6 |
|
nfcv |
⊢ Ⅎ 𝑤 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) |
7 |
|
nfsbc1v |
⊢ Ⅎ 𝑤 [ ( 2nd ‘ 𝑥 ) / 𝑤 ] 𝜑 |
8 |
6 7
|
nfsbcw |
⊢ Ⅎ 𝑤 [ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) / 𝑧 ] [ ( 2nd ‘ 𝑥 ) / 𝑤 ] 𝜑 |
9 |
5 8
|
nfsbcw |
⊢ Ⅎ 𝑤 [ ( 1st ‘ ( 1st ‘ 𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) / 𝑧 ] [ ( 2nd ‘ 𝑥 ) / 𝑤 ] 𝜑 |
10 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
11 |
|
sbcoteq1a |
⊢ ( 𝑥 = 〈 〈 𝑦 , 𝑧 〉 , 𝑤 〉 → ( [ ( 1st ‘ ( 1st ‘ 𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) / 𝑧 ] [ ( 2nd ‘ 𝑥 ) / 𝑤 ] 𝜑 ↔ 𝜑 ) ) |
12 |
1 4 9 10 11
|
ralxp3f |
⊢ ( ∀ 𝑥 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) [ ( 1st ‘ ( 1st ‘ 𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) / 𝑧 ] [ ( 2nd ‘ 𝑥 ) / 𝑤 ] 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐶 𝜑 ) |