Step |
Hyp |
Ref |
Expression |
1 |
|
infcl.1 |
⊢ ( 𝜑 → 𝑅 Or 𝐴 ) |
2 |
|
infcl.2 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) ) ) |
3 |
|
vex |
⊢ 𝑥 ∈ V |
4 |
|
vex |
⊢ 𝑦 ∈ V |
5 |
3 4
|
brcnv |
⊢ ( 𝑥 ◡ 𝑅 𝑦 ↔ 𝑦 𝑅 𝑥 ) |
6 |
5
|
bicomi |
⊢ ( 𝑦 𝑅 𝑥 ↔ 𝑥 ◡ 𝑅 𝑦 ) |
7 |
6
|
notbii |
⊢ ( ¬ 𝑦 𝑅 𝑥 ↔ ¬ 𝑥 ◡ 𝑅 𝑦 ) |
8 |
7
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ) |
9 |
4 3
|
brcnv |
⊢ ( 𝑦 ◡ 𝑅 𝑥 ↔ 𝑥 𝑅 𝑦 ) |
10 |
9
|
bicomi |
⊢ ( 𝑥 𝑅 𝑦 ↔ 𝑦 ◡ 𝑅 𝑥 ) |
11 |
|
vex |
⊢ 𝑧 ∈ V |
12 |
4 11
|
brcnv |
⊢ ( 𝑦 ◡ 𝑅 𝑧 ↔ 𝑧 𝑅 𝑦 ) |
13 |
12
|
bicomi |
⊢ ( 𝑧 𝑅 𝑦 ↔ 𝑦 ◡ 𝑅 𝑧 ) |
14 |
13
|
rexbii |
⊢ ( ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ↔ ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) |
15 |
10 14
|
imbi12i |
⊢ ( ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) ↔ ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) |
16 |
15
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) |
17 |
8 16
|
anbi12i |
⊢ ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) ) |
18 |
17
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) ) ↔ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) ) |
19 |
2 18
|
sylib |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) ) |