theorem ifeq3a (a b c: nat) (p: wff): $ (~p -> b = c) -> if p a b = if p a c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ifpos |
p -> if p a b = a |
| 2 |
|
ifpos |
p -> if p a c = a |
| 3 |
1, 2 |
eqtr4d |
p -> if p a b = if p a c |
| 4 |
3 |
a1i |
(~p -> b = c) -> p -> if p a b = if p a c |
| 5 |
|
ifeq3 |
b = c -> if p a b = if p a c |
| 6 |
5 |
imim2i |
(~p -> b = c) -> ~p -> if p a b = if p a c |
| 7 |
4, 6 |
casesd |
(~p -> b = c) -> if p a b = if p a c |
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,
the0)