Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → ( 2nd ‘ 𝐴 ) = ( 2nd ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) ) |
2 |
|
ot3rdg |
⊢ ( 𝑧 ∈ V → ( 2nd ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) = 𝑧 ) |
3 |
2
|
elv |
⊢ ( 2nd ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) = 𝑧 |
4 |
1 3
|
eqtr2di |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → 𝑧 = ( 2nd ‘ 𝐴 ) ) |
5 |
|
sbceq1a |
⊢ ( 𝑧 = ( 2nd ‘ 𝐴 ) → ( 𝜑 ↔ [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ) ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → ( 𝜑 ↔ [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ) ) |
7 |
|
2fveq3 |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = ( 2nd ‘ ( 1st ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) ) ) |
8 |
|
vex |
⊢ 𝑥 ∈ V |
9 |
|
vex |
⊢ 𝑦 ∈ V |
10 |
|
vex |
⊢ 𝑧 ∈ V |
11 |
|
ot2ndg |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 2nd ‘ ( 1st ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) ) = 𝑦 ) |
12 |
8 9 10 11
|
mp3an |
⊢ ( 2nd ‘ ( 1st ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) ) = 𝑦 |
13 |
7 12
|
eqtr2di |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → 𝑦 = ( 2nd ‘ ( 1st ‘ 𝐴 ) ) ) |
14 |
|
sbceq1a |
⊢ ( 𝑦 = ( 2nd ‘ ( 1st ‘ 𝐴 ) ) → ( [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ↔ [ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) / 𝑦 ] [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ) ) |
15 |
13 14
|
syl |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → ( [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ↔ [ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) / 𝑦 ] [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ) ) |
16 |
|
2fveq3 |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → ( 1st ‘ ( 1st ‘ 𝐴 ) ) = ( 1st ‘ ( 1st ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) ) ) |
17 |
|
ot1stg |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 1st ‘ ( 1st ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) ) = 𝑥 ) |
18 |
8 9 10 17
|
mp3an |
⊢ ( 1st ‘ ( 1st ‘ 〈 𝑥 , 𝑦 , 𝑧 〉 ) ) = 𝑥 |
19 |
16 18
|
eqtr2di |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → 𝑥 = ( 1st ‘ ( 1st ‘ 𝐴 ) ) ) |
20 |
|
sbceq1a |
⊢ ( 𝑥 = ( 1st ‘ ( 1st ‘ 𝐴 ) ) → ( [ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) / 𝑦 ] [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ↔ [ ( 1st ‘ ( 1st ‘ 𝐴 ) ) / 𝑥 ] [ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) / 𝑦 ] [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ) ) |
21 |
19 20
|
syl |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → ( [ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) / 𝑦 ] [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ↔ [ ( 1st ‘ ( 1st ‘ 𝐴 ) ) / 𝑥 ] [ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) / 𝑦 ] [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ) ) |
22 |
6 15 21
|
3bitrrd |
⊢ ( 𝐴 = 〈 𝑥 , 𝑦 , 𝑧 〉 → ( [ ( 1st ‘ ( 1st ‘ 𝐴 ) ) / 𝑥 ] [ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) / 𝑦 ] [ ( 2nd ‘ 𝐴 ) / 𝑧 ] 𝜑 ↔ 𝜑 ) ) |