Step |
Hyp |
Ref |
Expression |
1 |
|
supmo.1 |
⊢ ( 𝜑 → 𝑅 Or 𝐴 ) |
2 |
|
ancom |
⊢ ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) ↔ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ) ) |
3 |
2
|
anbi2ci |
⊢ ( ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) ↔ ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ) ) ) |
4 |
|
an42 |
⊢ ( ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) ↔ ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) ) |
5 |
|
an42 |
⊢ ( ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) ) ↔ ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ) ) ) |
6 |
3 4 5
|
3bitr4i |
⊢ ( ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) ↔ ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) ) ) |
7 |
|
ralnex |
⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ↔ ¬ ∃ 𝑦 ∈ 𝐵 𝑥 𝑅 𝑦 ) |
8 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 𝑅 𝑤 ↔ 𝑥 𝑅 𝑤 ) ) |
9 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 𝑅 𝑧 ↔ 𝑥 𝑅 𝑧 ) ) |
10 |
9
|
rexbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ↔ ∃ 𝑧 ∈ 𝐵 𝑥 𝑅 𝑧 ) ) |
11 |
8 10
|
imbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ↔ ( 𝑥 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑥 𝑅 𝑧 ) ) ) |
12 |
11
|
rspcva |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ( 𝑥 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑥 𝑅 𝑧 ) ) |
13 |
|
breq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑥 𝑅 𝑦 ↔ 𝑥 𝑅 𝑧 ) ) |
14 |
13
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ 𝐵 𝑥 𝑅 𝑦 ↔ ∃ 𝑧 ∈ 𝐵 𝑥 𝑅 𝑧 ) |
15 |
12 14
|
syl6ibr |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ( 𝑥 𝑅 𝑤 → ∃ 𝑦 ∈ 𝐵 𝑥 𝑅 𝑦 ) ) |
16 |
15
|
con3d |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ( ¬ ∃ 𝑦 ∈ 𝐵 𝑥 𝑅 𝑦 → ¬ 𝑥 𝑅 𝑤 ) ) |
17 |
7 16
|
syl5bi |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 → ¬ 𝑥 𝑅 𝑤 ) ) |
18 |
17
|
expimpd |
⊢ ( 𝑥 ∈ 𝐴 → ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ) → ¬ 𝑥 𝑅 𝑤 ) ) |
19 |
18
|
ad2antrl |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ) ) → ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ) → ¬ 𝑥 𝑅 𝑤 ) ) |
20 |
|
ralnex |
⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ↔ ¬ ∃ 𝑦 ∈ 𝐵 𝑤 𝑅 𝑦 ) |
21 |
|
breq1 |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 𝑅 𝑥 ↔ 𝑤 𝑅 𝑥 ) ) |
22 |
|
breq1 |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 𝑅 𝑧 ↔ 𝑤 𝑅 𝑧 ) ) |
23 |
22
|
rexbidv |
⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ↔ ∃ 𝑧 ∈ 𝐵 𝑤 𝑅 𝑧 ) ) |
24 |
21 23
|
imbi12d |
⊢ ( 𝑦 = 𝑤 → ( ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ↔ ( 𝑤 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑤 𝑅 𝑧 ) ) ) |
25 |
24
|
rspcva |
⊢ ( ( 𝑤 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ( 𝑤 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑤 𝑅 𝑧 ) ) |
26 |
|
breq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑤 𝑅 𝑦 ↔ 𝑤 𝑅 𝑧 ) ) |
27 |
26
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ 𝐵 𝑤 𝑅 𝑦 ↔ ∃ 𝑧 ∈ 𝐵 𝑤 𝑅 𝑧 ) |
28 |
25 27
|
syl6ibr |
⊢ ( ( 𝑤 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ( 𝑤 𝑅 𝑥 → ∃ 𝑦 ∈ 𝐵 𝑤 𝑅 𝑦 ) ) |
29 |
28
|
con3d |
⊢ ( ( 𝑤 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ( ¬ ∃ 𝑦 ∈ 𝐵 𝑤 𝑅 𝑦 → ¬ 𝑤 𝑅 𝑥 ) ) |
30 |
20 29
|
syl5bi |
⊢ ( ( 𝑤 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 → ¬ 𝑤 𝑅 𝑥 ) ) |
31 |
30
|
expimpd |
⊢ ( 𝑤 ∈ 𝐴 → ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) → ¬ 𝑤 𝑅 𝑥 ) ) |
32 |
31
|
ad2antll |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ) ) → ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) → ¬ 𝑤 𝑅 𝑥 ) ) |
33 |
19 32
|
anim12d |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ) ) → ( ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) ) → ( ¬ 𝑥 𝑅 𝑤 ∧ ¬ 𝑤 𝑅 𝑥 ) ) ) |
34 |
6 33
|
syl5bi |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ) ) → ( ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) → ( ¬ 𝑥 𝑅 𝑤 ∧ ¬ 𝑤 𝑅 𝑥 ) ) ) |
35 |
|
sotrieq2 |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ) ) → ( 𝑥 = 𝑤 ↔ ( ¬ 𝑥 𝑅 𝑤 ∧ ¬ 𝑤 𝑅 𝑥 ) ) ) |
36 |
34 35
|
sylibrd |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ) ) → ( ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) → 𝑥 = 𝑤 ) ) |
37 |
36
|
ralrimivva |
⊢ ( 𝑅 Or 𝐴 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) → 𝑥 = 𝑤 ) ) |
38 |
1 37
|
syl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) → 𝑥 = 𝑤 ) ) |
39 |
|
breq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 𝑅 𝑦 ↔ 𝑤 𝑅 𝑦 ) ) |
40 |
39
|
notbid |
⊢ ( 𝑥 = 𝑤 → ( ¬ 𝑥 𝑅 𝑦 ↔ ¬ 𝑤 𝑅 𝑦 ) ) |
41 |
40
|
ralbidv |
⊢ ( 𝑥 = 𝑤 → ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ↔ ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ) ) |
42 |
|
breq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝑦 𝑅 𝑥 ↔ 𝑦 𝑅 𝑤 ) ) |
43 |
42
|
imbi1d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ↔ ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) |
44 |
43
|
ralbidv |
⊢ ( 𝑥 = 𝑤 → ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) |
45 |
41 44
|
anbi12d |
⊢ ( 𝑥 = 𝑤 → ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ↔ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) ) |
46 |
45
|
rmo4 |
⊢ ( ∃* 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( ( ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑤 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑤 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) → 𝑥 = 𝑤 ) ) |
47 |
38 46
|
sylibr |
⊢ ( 𝜑 → ∃* 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) |