Step | Hyp | Ref
| Expression |
1 | | mptbi12f.1 |
. . . . . . . 8
⊢
Ⅎ𝑥𝐴 |
2 | | mptbi12f.2 |
. . . . . . . 8
⊢
Ⅎ𝑥𝐵 |
3 | 1, 2 | nfeq 2776 |
. . . . . . 7
⊢
Ⅎ𝑥 𝐴 = 𝐵 |
4 | | eleq2 2690 |
. . . . . . 7
⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
5 | 3, 4 | alrimi 2082 |
. . . . . 6
⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
6 | | ax-5 1839 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
7 | 6 | alimi 1739 |
. . . . . 6
⊢
(∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑥∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
8 | 5, 7 | syl 17 |
. . . . 5
⊢ (𝐴 = 𝐵 → ∀𝑥∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
9 | | eqeq2 2633 |
. . . . . . . . 9
⊢ (𝐷 = 𝐸 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) |
10 | 9 | alrimiv 1855 |
. . . . . . . 8
⊢ (𝐷 = 𝐸 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) |
11 | 10 | ralimi 2952 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐴 𝐷 = 𝐸 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) |
12 | | df-ral 2917 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐴 ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) |
13 | 11, 12 | sylib 208 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 𝐷 = 𝐸 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) |
14 | | 19.21v 1868 |
. . . . . . 7
⊢
(∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) |
15 | 14 | albii 1747 |
. . . . . 6
⊢
(∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) |
16 | 13, 15 | sylibr 224 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 𝐷 = 𝐸 → ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) |
17 | | id 22 |
. . . . . . 7
⊢ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))) |
18 | 17 | alanimi 1744 |
. . . . . 6
⊢
((∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))) |
19 | 18 | alanimi 1744 |
. . . . 5
⊢
((∀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)))) |
20 | 8, 16, 19 | syl2an 494 |
. . . 4
⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))) |
21 | | tsan2 33949 |
. . . . . . . . . . . 12
⊢ (¬
(((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 | 21 | ord 392 |
. . . . . . . . . . 11
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷))) |
23 | | tsbi2 33941 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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 | ord 392 |
. . . . . . . . . . . . . 14
⊢ (¬
(((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)))) |
25 | 24 | a1dd 50 |
. . . . . . . . . . . . 13
⊢ (¬
(((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 | | ax-1 6 |
. . . . . . . . . . . . 13
⊢ (¬
(((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))))) |
27 | 25, 26 | contrd 33899 |
. . . . . . . . . . . 12
⊢ (¬
(((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))) |
28 | 27 | a1d 25 |
. . . . . . . . . . 11
⊢ (¬
(((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)))) |
29 | 22, 28 | cnf1dd 33892 |
. . . . . . . . . 10
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
30 | | simplim 163 |
. . . . . . . . . . . . . 14
⊢ (¬
(((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)))) |
31 | 30 | a1d 25 |
. . . . . . . . . . . . 13
⊢ (¬
(((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))))) |
32 | | tsbi3 33942 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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 | 32 | ord 392 |
. . . . . . . . . . . . . 14
⊢ (¬
(((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))) |
34 | | tsan2 33949 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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))))) |
35 | 34 | a1d 25 |
. . . . . . . . . . . . . 14
⊢ (¬
(((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)))))) |
36 | 33, 35 | cnf1dd 33892 |
. . . . . . . . . . . . 13
⊢ (¬
(((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))))) |
37 | 31, 36 | contrd 33899 |
. . . . . . . . . . . 12
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵)) |
38 | 37 | ord 392 |
. . . . . . . . . . 11
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
39 | | tsan2 33949 |
. . . . . . . . . . . 12
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑥 ∈ 𝐵 ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
40 | 39 | a1d 25 |
. . . . . . . . . . 11
⊢ (¬
(((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)))) |
41 | 38, 40 | cnf1dd 33892 |
. . . . . . . . . 10
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
42 | 29, 41 | contrd 33899 |
. . . . . . . . 9
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → 𝑥 ∈ 𝐴) |
43 | 42 | a1d 25 |
. . . . . . . 8
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑥 ∈ 𝐴)) |
44 | 30 | a1d 25 |
. . . . . . . . 9
⊢ (¬
(((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))))) |
45 | | tsan3 33950 |
. . . . . . . . . 10
⊢ (¬
(((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 | 45 | a1d 25 |
. . . . . . . . 9
⊢ (¬
(((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)))))) |
47 | 44, 46 | cnfn2dd 33895 |
. . . . . . . 8
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))) |
48 | 43, 47 | mpdd 43 |
. . . . . . 7
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) |
49 | | notnotr 125 |
. . . . . . . . . . . . . . . 16
⊢ (¬
¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) |
50 | 49 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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))) |
51 | 39 | a1d 25 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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)))) |
52 | 50, 51 | cnfn2dd 33895 |
. . . . . . . . . . . . . 14
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → 𝑥 ∈ 𝐵)) |
53 | 37 | a1d 25 |
. . . . . . . . . . . . . 14
⊢ (¬
(((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))) |
54 | 52, 53 | cnfn2dd 33895 |
. . . . . . . . . . . . 13
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → 𝑥 ∈ 𝐴)) |
55 | | tsan3 33950 |
. . . . . . . . . . . . . . . 16
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑦 = 𝐸 ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
56 | 55 | a1d 25 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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)))) |
57 | 50, 56 | cnfn2dd 33895 |
. . . . . . . . . . . . . 14
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → 𝑦 = 𝐸)) |
58 | 30 | a1d 25 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
(((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))))) |
59 | 45 | a1d 25 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
(((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)))))) |
60 | 58, 59 | cnfn2dd 33895 |
. . . . . . . . . . . . . . . 16
⊢ (¬
(((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)))) |
61 | 54, 60 | mpdd 43 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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))) |
62 | | tsbi3 33942 |
. . . . . . . . . . . . . . . 16
⊢ (¬
(((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))) |
63 | 62 | a1d 25 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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)))) |
64 | 61, 63 | cnfn2dd 33895 |
. . . . . . . . . . . . . 14
⊢ (¬
(((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))) |
65 | 57, 64 | cnfn2dd 33895 |
. . . . . . . . . . . . 13
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → 𝑦 = 𝐷)) |
66 | 54, 65 | jcad 555 |
. . . . . . . . . . . 12
⊢ (¬
(((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))) |
67 | | ax-1 6 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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))))) |
68 | | tsim3 33939 |
. . . . . . . . . . . . . . . 16
⊢ (¬
(((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))))) |
69 | 68 | a1d 25 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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)))))) |
70 | 67, 69 | cnf2dd 33893 |
. . . . . . . . . . . . . 14
⊢ (¬
(((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)))) |
71 | | tsbi1 33940 |
. . . . . . . . . . . . . . 15
⊢ (¬
(((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)))) |
72 | 71 | a1d 25 |
. . . . . . . . . . . . . 14
⊢ (¬
(((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))))) |
73 | 70, 72 | cnf2dd 33893 |
. . . . . . . . . . . . 13
⊢ (¬
(((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)))) |
74 | 50, 73 | cnfn2dd 33895 |
. . . . . . . . . . . 12
⊢ (¬
(((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))) |
75 | 66, 74 | contrd 33899 |
. . . . . . . . . . 11
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) |
76 | 75 | a1d 25 |
. . . . . . . . . 10
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
77 | 27 | a1d 25 |
. . . . . . . . . 10
⊢ (¬
(((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)))) |
78 | 76, 77 | cnf2dd 33893 |
. . . . . . . . 9
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷))) |
79 | | tsan3 33950 |
. . . . . . . . . 10
⊢ (¬
(((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 | 79 | a1d 25 |
. . . . . . . . 9
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑦 = 𝐷 ∨ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))) |
81 | 78, 80 | cnfn2dd 33895 |
. . . . . . . 8
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑦 = 𝐷)) |
82 | 34 | a1d 25 |
. . . . . . . . . . . . 13
⊢ (¬
(((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)))))) |
83 | 44, 82 | cnfn2dd 33895 |
. . . . . . . . . . . 12
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
84 | | tsbi4 33943 |
. . . . . . . . . . . . 13
⊢ (¬
(((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 | 84 | a1d 25 |
. . . . . . . . . . . 12
⊢ (¬
(((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)))) |
86 | 83, 85 | cnfn2dd 33895 |
. . . . . . . . . . 11
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))) |
87 | 43, 86 | cnfn1dd 33894 |
. . . . . . . . . 10
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑥 ∈ 𝐵)) |
88 | | tsan1 33948 |
. . . . . . . . . . . 12
⊢ (¬
(((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 | 88 | a1d 25 |
. . . . . . . . . . 11
⊢ (¬
(((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)))) |
90 | 76, 89 | cnf2dd 33893 |
. . . . . . . . . 10
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑦 = 𝐸))) |
91 | 87, 90 | cnfn1dd 33894 |
. . . . . . . . 9
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ 𝑦 = 𝐸)) |
92 | | tsbi4 33943 |
. . . . . . . . . . 11
⊢ (¬
(((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 | a1d 25 |
. . . . . . . . . 10
⊢ (¬
(((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)))) |
94 | 93 | or32dd 33896 |
. . . . . . . . 9
⊢ (¬
(((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))) |
95 | 91, 94 | cnf2dd 33893 |
. . . . . . . 8
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑦 = 𝐷 ∨ ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))) |
96 | 81, 95 | cnfn1dd 33894 |
. . . . . . 7
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) |
97 | 48, 96 | contrd 33899 |
. . . . . 6
⊢ (¬
(((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ⊥) |
98 | 97 | efald2 33877 |
. . . . 5
⊢ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
99 | 98 | 2alimi 1740 |
. . . 4
⊢
(∀𝑥∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
100 | 20, 99 | syl 17 |
. . 3
⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
101 | | eqopab2b 5005 |
. . 3
⊢
({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)} ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) |
102 | 100, 101 | sylibr 224 |
. 2
⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)}) |
103 | | df-mpt 4730 |
. 2
⊢ (𝑥 ∈ 𝐴 ↦ 𝐷) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)} |
104 | | df-mpt 4730 |
. 2
⊢ (𝑥 ∈ 𝐵 ↦ 𝐸) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)} |
105 | 102, 103,
104 | 3eqtr4g 2681 |
1
⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐵 ↦ 𝐸)) |