theorem resres (A B F: set): $ F |` A |` B == F |` A i^i B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqstr4 |
F |` A |` B == F i^i (Xp A _V i^i Xp B _V) -> F |` A i^i B == F i^i (Xp A _V i^i Xp B _V) -> F |` A |` B == F |` A i^i B |
| 2 |
|
inass |
F i^i Xp A _V i^i Xp B _V == F i^i (Xp A _V i^i Xp B _V) |
| 3 |
2 |
conv res |
F |` A |` B == F i^i (Xp A _V i^i Xp B _V) |
| 4 |
1, 3 |
ax_mp |
F |` A i^i B == F i^i (Xp A _V i^i Xp B _V) -> F |` A |` B == F |` A i^i B |
| 5 |
|
ineq2 |
Xp (A i^i B) _V == Xp A _V i^i Xp B _V -> F i^i Xp (A i^i B) _V == F i^i (Xp A _V i^i Xp B _V) |
| 6 |
5 |
conv res |
Xp (A i^i B) _V == Xp A _V i^i Xp B _V -> F |` A i^i B == F i^i (Xp A _V i^i Xp B _V) |
| 7 |
|
xpindi |
Xp (A i^i B) _V == Xp A _V i^i Xp B _V |
| 8 |
6, 7 |
ax_mp |
F |` A i^i B == F i^i (Xp A _V i^i Xp B _V) |
| 9 |
4, 8 |
ax_mp |
F |` A |` B == F |` A i^i B |
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),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)