Step |
Hyp |
Ref |
Expression |
1 |
|
mptbi12f.1 |
⊢ Ⅎ 𝑥 𝐴 |
2 |
|
mptbi12f.2 |
⊢ Ⅎ 𝑥 𝐵 |
3 |
1 2
|
nfeq |
⊢ Ⅎ 𝑥 𝐴 = 𝐵 |
4 |
|
eleq2 |
⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
5 |
3 4
|
alrimi |
⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
6 |
|
ax-5 |
⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ∀ 𝑦 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
7 |
5 6
|
sylg |
⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∀ 𝑦 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
8 |
|
eqeq2 |
⊢ ( 𝐷 = 𝐸 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) |
9 |
8
|
alrimiv |
⊢ ( 𝐷 = 𝐸 → ∀ 𝑦 ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) |
10 |
9
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) |
11 |
|
df-ral |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
12 |
10 11
|
sylib |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
13 |
|
19.21v |
⊢ ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
14 |
13
|
albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
15 |
12 14
|
sylibr |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀ 𝑥 ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
16 |
|
id |
⊢ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
17 |
16
|
alanimi |
⊢ ( ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
18 |
17
|
alanimi |
⊢ ( ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
19 |
7 15 18
|
syl2an |
⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝐷 = 𝐸 ) → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
20 |
|
tsan2 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( 𝑥 ∈ 𝐴 ∨ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) ) |
21 |
20
|
ord |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥 ∈ 𝐴 → ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) ) |
22 |
|
tsbi2 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ∨ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
23 |
22
|
ord |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
24 |
23
|
a1dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) → ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) ) |
25 |
|
ax-1 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) → ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) ) |
26 |
24 25
|
contrd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
27 |
26
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥 ∈ 𝐴 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
28 |
21 27
|
cnf1dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
29 |
|
simplim |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
30 |
29
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) |
31 |
|
tsbi3 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵 ) ∨ ¬ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
32 |
31
|
ord |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵 ) → ¬ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
33 |
|
tsan2 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∨ ¬ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) |
34 |
33
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∨ ¬ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) ) |
35 |
32 34
|
cnf1dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵 ) → ¬ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) |
36 |
30 35
|
contrd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵 ) ) |
37 |
36
|
ord |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵 ) ) |
38 |
|
tsan2 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( 𝑥 ∈ 𝐵 ∨ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
39 |
38
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∨ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
40 |
37 39
|
cnf1dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ 𝑥 ∈ 𝐴 → ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
41 |
28 40
|
contrd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → 𝑥 ∈ 𝐴 ) |
42 |
41
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → 𝑥 ∈ 𝐴 ) ) |
43 |
29
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) |
44 |
|
tsan3 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ∨ ¬ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) |
45 |
44
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ∨ ¬ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) ) |
46 |
43 45
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
47 |
42 46
|
mpdd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
48 |
|
notnotr |
⊢ ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) |
49 |
48
|
a1i |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
50 |
38
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑥 ∈ 𝐵 ∨ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
51 |
49 50
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → 𝑥 ∈ 𝐵 ) ) |
52 |
36
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵 ) ) ) |
53 |
51 52
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → 𝑥 ∈ 𝐴 ) ) |
54 |
|
tsan3 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( 𝑦 = 𝐸 ∨ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
55 |
54
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑦 = 𝐸 ∨ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
56 |
49 55
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → 𝑦 = 𝐸 ) ) |
57 |
29
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) |
58 |
44
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ∨ ¬ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) ) |
59 |
57 58
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
60 |
53 59
|
mpdd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
61 |
|
tsbi3 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( 𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸 ) ∨ ¬ ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
62 |
61
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( ( 𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸 ) ∨ ¬ ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
63 |
60 62
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸 ) ) ) |
64 |
56 63
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → 𝑦 = 𝐷 ) ) |
65 |
53 64
|
jcad |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) ) |
66 |
|
ax-1 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) ) |
67 |
|
tsim3 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ∨ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) ) |
68 |
67
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( ¬ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ∨ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) ) ) |
69 |
66 68
|
cnf2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ¬ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
70 |
|
tsbi1 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ∨ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
71 |
70
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( ( ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ∨ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) ) |
72 |
69 71
|
cnf2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ( ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
73 |
49 72
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) → ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) ) |
74 |
65 73
|
contrd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) |
75 |
74
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ¬ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
76 |
26
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
77 |
75 76
|
cnf2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) ) |
78 |
|
tsan3 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( 𝑦 = 𝐷 ∨ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) ) |
79 |
78
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑦 = 𝐷 ∨ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) ) ) |
80 |
77 79
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → 𝑦 = 𝐷 ) ) |
81 |
33
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∨ ¬ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) ) ) |
82 |
43 81
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
83 |
|
tsbi4 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( ¬ 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∨ ¬ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
84 |
83
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( ¬ 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∨ ¬ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) ) |
85 |
82 84
|
cnfn2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ¬ 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ) ) |
86 |
42 85
|
cnfn1dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → 𝑥 ∈ 𝐵 ) ) |
87 |
|
tsan1 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( ¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑦 = 𝐸 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
88 |
87
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( ¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑦 = 𝐸 ) ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) ) |
89 |
75 88
|
cnf2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑦 = 𝐸 ) ) ) |
90 |
86 89
|
cnfn1dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ¬ 𝑦 = 𝐸 ) ) |
91 |
|
tsbi4 |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ( ¬ 𝑦 = 𝐷 ∨ 𝑦 = 𝐸 ) ∨ ¬ ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
92 |
91
|
a1d |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( ¬ 𝑦 = 𝐷 ∨ 𝑦 = 𝐸 ) ∨ ¬ ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
93 |
92
|
or32dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ( ¬ 𝑦 = 𝐷 ∨ ¬ ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ∨ 𝑦 = 𝐸 ) ) ) |
94 |
90 93
|
cnf2dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ( ¬ 𝑦 = 𝐷 ∨ ¬ ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) ) |
95 |
80 94
|
cnfn1dd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ( ¬ ⊥ → ¬ ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) |
96 |
47 95
|
contrd |
⊢ ( ¬ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) → ⊥ ) |
97 |
96
|
efald2 |
⊢ ( ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
98 |
97
|
2alimi |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐷 ↔ 𝑦 = 𝐸 ) ) ) → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
99 |
19 98
|
syl |
⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝐷 = 𝐸 ) → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
100 |
|
eqopab2bw |
⊢ ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) } = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) } ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) ) ) |
101 |
99 100
|
sylibr |
⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝐷 = 𝐸 ) → { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) } = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) } ) |
102 |
|
df-mpt |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) } |
103 |
|
df-mpt |
⊢ ( 𝑥 ∈ 𝐵 ↦ 𝐸 ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸 ) } |
104 |
101 102 103
|
3eqtr4g |
⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝐷 = 𝐸 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐸 ) ) |