Proof of Theorem u3lem11a
Step | Hyp | Ref
| Expression |
1 | | ud3lem1 570 |
. . . . 5
((b →3 a) →3 (a →3 b)) = (b ∪
(b⊥ ∩ a⊥ )) |
2 | | ancom 74 |
. . . . . . . 8
(b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ ) |
3 | | anor3 90 |
. . . . . . . 8
(a⊥ ∩ b⊥ ) = (a ∪ b)⊥ |
4 | 2, 3 | ax-r2 36 |
. . . . . . 7
(b⊥ ∩ a⊥ ) = (a ∪ b)⊥ |
5 | 4 | lor 70 |
. . . . . 6
(b ∪ (b⊥ ∩ a⊥ )) = (b ∪ (a ∪
b)⊥ ) |
6 | | oran1 91 |
. . . . . 6
(b ∪ (a ∪ b)⊥ ) = (b⊥ ∩ (a ∪ b))⊥ |
7 | 5, 6 | ax-r2 36 |
. . . . 5
(b ∪ (b⊥ ∩ a⊥ )) = (b⊥ ∩ (a ∪ b))⊥ |
8 | 1, 7 | ax-r2 36 |
. . . 4
((b →3 a) →3 (a →3 b)) = (b⊥ ∩ (a ∪ b))⊥ |
9 | 8 | con2 67 |
. . 3
((b →3 a) →3 (a →3 b))⊥ = (b⊥ ∩ (a ∪ b)) |
10 | 9 | ud3lem0a 260 |
. 2
(a →3 ((b →3 a) →3 (a →3 b))⊥ ) = (a →3 (b⊥ ∩ (a ∪ b))) |
11 | | u3lem11 786 |
. 2
(a →3 (b⊥ ∩ (a ∪ b))) =
(a →3 b⊥ ) |
12 | 10, 11 | ax-r2 36 |
1
(a →3 ((b →3 a) →3 (a →3 b))⊥ ) = (a →3 b⊥ ) |