Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1021.1 |
⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
2 |
|
bnj1021.2 |
⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
3 |
|
bnj1021.3 |
⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
4 |
|
bnj1021.4 |
⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
5 |
|
bnj1021.5 |
⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) |
6 |
|
bnj1021.6 |
⊢ ( 𝜂 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ) |
7 |
|
bnj1021.13 |
⊢ 𝐷 = ( ω ∖ { ∅ } ) |
8 |
|
bnj1021.14 |
⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) } |
9 |
1 2 3 4 5 6 7 8
|
bnj996 |
⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) |
10 |
|
anclb |
⊢ ( ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) ) |
11 |
|
bnj252 |
⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( 𝜃 ∧ ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) |
12 |
11
|
imbi2i |
⊢ ( ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) ) |
13 |
10 12
|
bitr4i |
⊢ ( ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) |
14 |
13
|
2exbii |
⊢ ( ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) |
15 |
14
|
3exbii |
⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) |
16 |
9 15
|
mpbi |
⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) |
17 |
|
19.37v |
⊢ ( ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) |
18 |
|
bnj1019 |
⊢ ( ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) |
19 |
18
|
imbi2i |
⊢ ( ( 𝜃 → ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) ) |
20 |
17 19
|
bitri |
⊢ ( ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) ) |
21 |
20
|
2exbii |
⊢ ( ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑖 ∃ 𝑚 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) ) |
22 |
21
|
2exbii |
⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) ) |
23 |
16 22
|
mpbi |
⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) |