theorem allal (l: nat) {x: nat} (p: wff x):
$ all {x | p} l <-> A. x (x IN l -> p) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bicom |
(A. x (x IN l -> p) <-> all {x | p} l) -> (all {x | p} l <-> A. x (x IN l -> p)) |
| 2 |
|
ssab2 |
A. x (x e. lmems l -> p) <-> lmems l C_ {x | p} |
| 3 |
2 |
conv all, lmem |
A. x (x IN l -> p) <-> all {x | p} l |
| 4 |
1, 3 |
ax_mp |
all {x | p} l <-> A. x (x IN l -> p) |
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)