theorem ifeq (_p1 _p2: wff) (_a1 _a2 _b1 _b2: nat):
$ (_p1 <-> _p2) ->
_a1 = _a2 ->
_b1 = _b2 ->
if _p1 _a1 _b1 = if _p2 _a2 _b2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
anl |
(_p1 <-> _p2) /\ _a1 = _a2 -> (_p1 <-> _p2) |
| 2 |
1 |
anwl |
(_p1 <-> _p2) /\ _a1 = _a2 /\ _b1 = _b2 -> (_p1 <-> _p2) |
| 3 |
|
anr |
(_p1 <-> _p2) /\ _a1 = _a2 -> _a1 = _a2 |
| 4 |
3 |
anwl |
(_p1 <-> _p2) /\ _a1 = _a2 /\ _b1 = _b2 -> _a1 = _a2 |
| 5 |
|
anr |
(_p1 <-> _p2) /\ _a1 = _a2 /\ _b1 = _b2 -> _b1 = _b2 |
| 6 |
2, 4, 5 |
ifeqd |
(_p1 <-> _p2) /\ _a1 = _a2 /\ _b1 = _b2 -> if _p1 _a1 _b1 = if _p2 _a2 _b2 |
| 7 |
6 |
exp |
(_p1 <-> _p2) /\ _a1 = _a2 -> _b1 = _b2 -> if _p1 _a1 _b1 = if _p2 _a2 _b2 |
| 8 |
7 |
exp |
(_p1 <-> _p2) -> _a1 = _a2 -> _b1 = _b2 -> if _p1 _a1 _b1 = if _p2 _a2 _b2 |
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)