theorem notor (a b: wff): $ ~(a \/ b) <-> ~a /\ ~b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
con1b |
(~(~a /\ ~b) <-> a \/ b) -> (~(a \/ b) <-> ~a /\ ~b) |
| 2 |
|
bitr4 |
(~(~a /\ ~b) <-> ~~a \/ ~~b) -> (a \/ b <-> ~~a \/ ~~b) -> (~(~a /\ ~b) <-> a \/ b) |
| 3 |
|
notan |
~(~a /\ ~b) <-> ~~a \/ ~~b |
| 4 |
2, 3 |
ax_mp |
(a \/ b <-> ~~a \/ ~~b) -> (~(~a /\ ~b) <-> a \/ b) |
| 5 |
|
notnot |
a <-> ~~a |
| 6 |
|
notnot |
b <-> ~~b |
| 7 |
5, 6 |
oreqi |
a \/ b <-> ~~a \/ ~~b |
| 8 |
4, 7 |
ax_mp |
~(~a /\ ~b) <-> a \/ b |
| 9 |
1, 8 |
ax_mp |
~(a \/ b) <-> ~a /\ ~b |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp)