theorem andi (a b c: wff): $ a /\ (b \/ c) <-> a /\ b \/ a /\ c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ian |
a -> b -> a /\ b |
| 2 |
|
ian |
a -> c -> a /\ c |
| 3 |
1, 2 |
orimd |
a -> b \/ c -> a /\ b \/ a /\ c |
| 4 |
3 |
imp |
a /\ (b \/ c) -> a /\ b \/ a /\ c |
| 5 |
|
eor |
(a /\ b -> a /\ (b \/ c)) -> (a /\ c -> a /\ (b \/ c)) -> a /\ b \/ a /\ c -> a /\ (b \/ c) |
| 6 |
|
anim2 |
(b -> b \/ c) -> a /\ b -> a /\ (b \/ c) |
| 7 |
|
orl |
b -> b \/ c |
| 8 |
6, 7 |
ax_mp |
a /\ b -> a /\ (b \/ c) |
| 9 |
5, 8 |
ax_mp |
(a /\ c -> a /\ (b \/ c)) -> a /\ b \/ a /\ c -> a /\ (b \/ c) |
| 10 |
|
anim2 |
(c -> b \/ c) -> a /\ c -> a /\ (b \/ c) |
| 11 |
|
orr |
c -> b \/ c |
| 12 |
10, 11 |
ax_mp |
a /\ c -> a /\ (b \/ c) |
| 13 |
9, 12 |
ax_mp |
a /\ b \/ a /\ c -> a /\ (b \/ c) |
| 14 |
4, 13 |
ibii |
a /\ (b \/ c) <-> a /\ b \/ a /\ c |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp)