Step |
Hyp |
Ref |
Expression |
1 |
|
br6.1 |
⊢ ( 𝑎 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
br6.2 |
⊢ ( 𝑏 = 𝐵 → ( 𝜓 ↔ 𝜒 ) ) |
3 |
|
br6.3 |
⊢ ( 𝑐 = 𝐶 → ( 𝜒 ↔ 𝜃 ) ) |
4 |
|
br6.4 |
⊢ ( 𝑑 = 𝐷 → ( 𝜃 ↔ 𝜏 ) ) |
5 |
|
br6.5 |
⊢ ( 𝑒 = 𝐸 → ( 𝜏 ↔ 𝜂 ) ) |
6 |
|
br6.6 |
⊢ ( 𝑓 = 𝐹 → ( 𝜂 ↔ 𝜁 ) ) |
7 |
|
br6.7 |
⊢ ( 𝑥 = 𝑋 → 𝑃 = 𝑄 ) |
8 |
|
br6.8 |
⊢ 𝑅 = { ⟨ 𝑝 , 𝑞 ⟩ ∣ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) } |
9 |
|
opex |
⊢ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∈ V |
10 |
|
opex |
⊢ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∈ V |
11 |
|
eqeq1 |
⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ) ) |
12 |
|
eqcom |
⊢ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ↔ ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) |
13 |
11 12
|
bitrdi |
⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ↔ ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) ) |
14 |
13
|
3anbi1d |
⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) ) |
15 |
14
|
rexbidv |
⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) ) |
16 |
15
|
2rexbidv |
⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) ) |
17 |
16
|
2rexbidv |
⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) ) |
18 |
17
|
2rexbidv |
⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) ) |
19 |
|
eqeq1 |
⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ) ) |
20 |
|
eqcom |
⊢ ( ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ↔ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) |
21 |
19 20
|
bitrdi |
⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ↔ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) |
22 |
21
|
3anbi2d |
⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
23 |
22
|
rexbidv |
⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
24 |
23
|
2rexbidv |
⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
25 |
24
|
2rexbidv |
⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
26 |
25
|
2rexbidv |
⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
27 |
9 10 18 26 8
|
brab |
⊢ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ 𝑅 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) |
28 |
|
vex |
⊢ 𝑎 ∈ V |
29 |
|
opex |
⊢ ⟨ 𝑏 , 𝑐 ⟩ ∈ V |
30 |
28 29
|
opth |
⊢ ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ↔ ( 𝑎 = 𝐴 ∧ ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ ) ) |
31 |
|
vex |
⊢ 𝑏 ∈ V |
32 |
|
vex |
⊢ 𝑐 ∈ V |
33 |
31 32
|
opth |
⊢ ( ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝑏 = 𝐵 ∧ 𝑐 = 𝐶 ) ) |
34 |
2 3
|
sylan9bb |
⊢ ( ( 𝑏 = 𝐵 ∧ 𝑐 = 𝐶 ) → ( 𝜓 ↔ 𝜃 ) ) |
35 |
33 34
|
sylbi |
⊢ ( ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ → ( 𝜓 ↔ 𝜃 ) ) |
36 |
1 35
|
sylan9bb |
⊢ ( ( 𝑎 = 𝐴 ∧ ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ ) → ( 𝜑 ↔ 𝜃 ) ) |
37 |
30 36
|
sylbi |
⊢ ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( 𝜑 ↔ 𝜃 ) ) |
38 |
|
vex |
⊢ 𝑑 ∈ V |
39 |
|
opex |
⊢ ⟨ 𝑒 , 𝑓 ⟩ ∈ V |
40 |
38 39
|
opth |
⊢ ( ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ( 𝑑 = 𝐷 ∧ ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ ) ) |
41 |
|
vex |
⊢ 𝑒 ∈ V |
42 |
|
vex |
⊢ 𝑓 ∈ V |
43 |
41 42
|
opth |
⊢ ( ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ ↔ ( 𝑒 = 𝐸 ∧ 𝑓 = 𝐹 ) ) |
44 |
5 6
|
sylan9bb |
⊢ ( ( 𝑒 = 𝐸 ∧ 𝑓 = 𝐹 ) → ( 𝜏 ↔ 𝜁 ) ) |
45 |
43 44
|
sylbi |
⊢ ( ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ → ( 𝜏 ↔ 𝜁 ) ) |
46 |
4 45
|
sylan9bb |
⊢ ( ( 𝑑 = 𝐷 ∧ ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ ) → ( 𝜃 ↔ 𝜁 ) ) |
47 |
40 46
|
sylbi |
⊢ ( ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( 𝜃 ↔ 𝜁 ) ) |
48 |
37 47
|
sylan9bb |
⊢ ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) → ( 𝜑 ↔ 𝜁 ) ) |
49 |
48
|
biimp3a |
⊢ ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) |
50 |
49
|
a1i |
⊢ ( ( ( ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃 ) ) ∧ ( 𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃 ) ) ∧ ( 𝑑 ∈ 𝑃 ∧ 𝑒 ∈ 𝑃 ) ) ∧ 𝑓 ∈ 𝑃 ) → ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) ) |
51 |
50
|
rexlimdva |
⊢ ( ( ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃 ) ) ∧ ( 𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃 ) ) ∧ ( 𝑑 ∈ 𝑃 ∧ 𝑒 ∈ 𝑃 ) ) → ( ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) ) |
52 |
51
|
rexlimdvva |
⊢ ( ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃 ) ) ∧ ( 𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃 ) ) → ( ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) ) |
53 |
52
|
rexlimdvva |
⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃 ) ) → ( ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) ) |
54 |
53
|
rexlimdvva |
⊢ ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) ) |
55 |
|
simpl1 |
⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → 𝑋 ∈ 𝑆 ) |
56 |
|
simpl2 |
⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ) |
57 |
|
opeq1 |
⊢ ( 𝑑 = 𝐷 → ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ) |
58 |
57
|
eqeq1d |
⊢ ( 𝑑 = 𝐷 → ( ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) |
59 |
58 4
|
3anbi23d |
⊢ ( 𝑑 = 𝐷 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜏 ) ) ) |
60 |
|
opeq1 |
⊢ ( 𝑒 = 𝐸 → ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝑓 ⟩ ) |
61 |
60
|
opeq2d |
⊢ ( 𝑒 = 𝐸 → ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) |
62 |
61
|
eqeq1d |
⊢ ( 𝑒 = 𝐸 → ( ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) |
63 |
62 5
|
3anbi23d |
⊢ ( 𝑒 = 𝐸 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜏 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜂 ) ) ) |
64 |
|
opeq2 |
⊢ ( 𝑓 = 𝐹 → ⟨ 𝐸 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ ) |
65 |
64
|
opeq2d |
⊢ ( 𝑓 = 𝐹 → ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) |
66 |
65
|
eqeq1d |
⊢ ( 𝑓 = 𝐹 → ( ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) |
67 |
66 6
|
3anbi23d |
⊢ ( 𝑓 = 𝐹 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜂 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜁 ) ) ) |
68 |
|
eqid |
⊢ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ |
69 |
|
eqid |
⊢ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ |
70 |
68 69
|
pm3.2i |
⊢ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) |
71 |
|
df-3an |
⊢ ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜁 ) ↔ ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ 𝜁 ) ) |
72 |
70 71
|
mpbiran |
⊢ ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜁 ) ↔ 𝜁 ) |
73 |
67 72
|
bitrdi |
⊢ ( 𝑓 = 𝐹 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜂 ) ↔ 𝜁 ) ) |
74 |
59 63 73
|
rspc3ev |
⊢ ( ( ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ∧ 𝜁 ) → ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) |
75 |
74
|
3ad2antl3 |
⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) |
76 |
|
opeq1 |
⊢ ( 𝑎 = 𝐴 → ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ) |
77 |
76
|
eqeq1d |
⊢ ( 𝑎 = 𝐴 → ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) ) |
78 |
77 1
|
3anbi13d |
⊢ ( 𝑎 = 𝐴 → ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ) ) |
79 |
78
|
rexbidv |
⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ) ) |
80 |
79
|
2rexbidv |
⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ) ) |
81 |
|
opeq1 |
⊢ ( 𝑏 = 𝐵 → ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝑐 ⟩ ) |
82 |
81
|
opeq2d |
⊢ ( 𝑏 = 𝐵 → ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ ) |
83 |
82
|
eqeq1d |
⊢ ( 𝑏 = 𝐵 → ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) ) |
84 |
83 2
|
3anbi13d |
⊢ ( 𝑏 = 𝐵 → ( ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ) ) |
85 |
84
|
rexbidv |
⊢ ( 𝑏 = 𝐵 → ( ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ↔ ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ) ) |
86 |
85
|
2rexbidv |
⊢ ( 𝑏 = 𝐵 → ( ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ↔ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ) ) |
87 |
|
opeq2 |
⊢ ( 𝑐 = 𝐶 → ⟨ 𝐵 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ ) |
88 |
87
|
opeq2d |
⊢ ( 𝑐 = 𝐶 → ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) |
89 |
88
|
eqeq1d |
⊢ ( 𝑐 = 𝐶 → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) ) |
90 |
89 3
|
3anbi13d |
⊢ ( 𝑐 = 𝐶 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) ) |
91 |
90
|
rexbidv |
⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ↔ ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) ) |
92 |
91
|
2rexbidv |
⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ↔ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) ) |
93 |
80 86 92
|
rspc3ev |
⊢ ( ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) → ∃ 𝑎 ∈ 𝑄 ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) |
94 |
56 75 93
|
syl2anc |
⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → ∃ 𝑎 ∈ 𝑄 ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) |
95 |
7
|
rexeqdv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
96 |
7 95
|
rexeqbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
97 |
7 96
|
rexeqbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
98 |
7 97
|
rexeqbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
99 |
7 98
|
rexeqbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
100 |
7 99
|
rexeqbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑎 ∈ 𝑄 ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
101 |
100
|
rspcev |
⊢ ( ( 𝑋 ∈ 𝑆 ∧ ∃ 𝑎 ∈ 𝑄 ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) → ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) |
102 |
55 94 101
|
syl2anc |
⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) |
103 |
102
|
ex |
⊢ ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( 𝜁 → ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) ) |
104 |
54 103
|
impbid |
⊢ ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ 𝜁 ) ) |
105 |
27 104
|
bitrid |
⊢ ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ 𝑅 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ 𝜁 ) ) |