theorem ifT (A: set) (a b: nat) (p: wff): $ a e. A /\ b e. A -> if p a b e. A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eleq1 |
if p a b = a -> (if p a b e. A <-> a e. A) |
| 2 |
|
eleq1 |
if p a b = b -> (if p a b e. A <-> b e. A) |
| 3 |
|
anll |
a e. A /\ b e. A /\ p -> a e. A |
| 4 |
|
anlr |
a e. A /\ b e. A /\ ~p -> b e. A |
| 5 |
1, 2, 3, 4 |
ifbothd |
a e. A /\ b e. A -> if p a b e. A |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp,
itru),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7,
ax_10,
ax_11,
ax_12),
axs_set
(elab,
ax_8),
axs_the
(theid)