theorem oriman (a b: wff): $ a <-> b <-> a \/ b -> a /\ b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
anr |
(a <-> b) /\ a -> a |
| 2 |
|
bi1 |
(a <-> b) -> a -> b |
| 3 |
2 |
imp |
(a <-> b) /\ a -> b |
| 4 |
1, 3 |
iand |
(a <-> b) /\ a -> a /\ b |
| 5 |
|
bi2 |
(a <-> b) -> b -> a |
| 6 |
5 |
imp |
(a <-> b) /\ b -> a |
| 7 |
|
anr |
(a <-> b) /\ b -> b |
| 8 |
6, 7 |
iand |
(a <-> b) /\ b -> a /\ b |
| 9 |
4, 8 |
eorda |
(a <-> b) -> a \/ b -> a /\ b |
| 10 |
|
orl |
a -> a \/ b |
| 11 |
10 |
imim1i |
(a \/ b -> a /\ b) -> a -> a /\ b |
| 12 |
11 |
imp |
(a \/ b -> a /\ b) /\ a -> a /\ b |
| 13 |
12 |
anrd |
(a \/ b -> a /\ b) /\ a -> b |
| 14 |
|
orr |
b -> a \/ b |
| 15 |
14 |
imim1i |
(a \/ b -> a /\ b) -> b -> a /\ b |
| 16 |
15 |
imp |
(a \/ b -> a /\ b) /\ b -> a /\ b |
| 17 |
16 |
anld |
(a \/ b -> a /\ b) /\ b -> a |
| 18 |
13, 17 |
ibida |
(a \/ b -> a /\ b) -> (a <-> b) |
| 19 |
9, 18 |
ibii |
a <-> b <-> a \/ b -> a /\ b |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp)