Proof of Theorem vtocl2gaf
Step | Hyp | Ref
| Expression |
1 | | vtocl2gaf.a |
. . 3
⊢
ℲxA |
2 | | vtocl2gaf.b |
. . 3
⊢
ℲyA |
3 | | vtocl2gaf.c |
. . 3
⊢
ℲyB |
4 | 1 | nfel1 2499 |
. . . . 5
⊢ Ⅎx A ∈ C |
5 | | nfv 1619 |
. . . . 5
⊢ Ⅎx y ∈ D |
6 | 4, 5 | nfan 1824 |
. . . 4
⊢ Ⅎx(A ∈ C ∧ y ∈ D) |
7 | | vtocl2gaf.1 |
. . . 4
⊢ Ⅎxψ |
8 | 6, 7 | nfim 1813 |
. . 3
⊢ Ⅎx((A ∈ C ∧ y ∈ D) →
ψ) |
9 | 2 | nfel1 2499 |
. . . . 5
⊢ Ⅎy A ∈ C |
10 | 3 | nfel1 2499 |
. . . . 5
⊢ Ⅎy B ∈ D |
11 | 9, 10 | nfan 1824 |
. . . 4
⊢ Ⅎy(A ∈ C ∧ B ∈ D) |
12 | | vtocl2gaf.2 |
. . . 4
⊢ Ⅎyχ |
13 | 11, 12 | nfim 1813 |
. . 3
⊢ Ⅎy((A ∈ C ∧ B ∈ D) →
χ) |
14 | | eleq1 2413 |
. . . . 5
⊢ (x = A →
(x ∈
C ↔ A ∈ C)) |
15 | 14 | anbi1d 685 |
. . . 4
⊢ (x = A →
((x ∈
C ∧
y ∈
D) ↔ (A ∈ C ∧ y ∈ D))) |
16 | | vtocl2gaf.3 |
. . . 4
⊢ (x = A →
(φ ↔ ψ)) |
17 | 15, 16 | imbi12d 311 |
. . 3
⊢ (x = A →
(((x ∈
C ∧
y ∈
D) → φ) ↔ ((A ∈ C ∧ y ∈ D) → ψ))) |
18 | | eleq1 2413 |
. . . . 5
⊢ (y = B →
(y ∈
D ↔ B ∈ D)) |
19 | 18 | anbi2d 684 |
. . . 4
⊢ (y = B →
((A ∈
C ∧
y ∈
D) ↔ (A ∈ C ∧ B ∈ D))) |
20 | | vtocl2gaf.4 |
. . . 4
⊢ (y = B →
(ψ ↔ χ)) |
21 | 19, 20 | imbi12d 311 |
. . 3
⊢ (y = B →
(((A ∈
C ∧
y ∈
D) → ψ) ↔ ((A ∈ C ∧ B ∈ D) → χ))) |
22 | | vtocl2gaf.5 |
. . 3
⊢ ((x ∈ C ∧ y ∈ D) → φ) |
23 | 1, 2, 3, 8, 13, 17, 21, 22 | vtocl2gf 2916 |
. 2
⊢ ((A ∈ C ∧ B ∈ D) → ((A
∈ C ∧ B ∈ D) →
χ)) |
24 | 23 | pm2.43i 43 |
1
⊢ ((A ∈ C ∧ B ∈ D) → χ) |