Theorem prellams ≪ | index | src | ≫ 

theorem prellams (a b: nat) {x: nat} (v: nat x):
  $ a, b e. \ x, v <-> b = N[a / x] v $;
StepHypRefExpression
1 bitr 
(a, b e. \ x, v <-> E. x a, b = x, v) -> (E. x a, b = x, v <-> b = N[a / x] v) -> (a, b e. \ x, v <-> b = N[a / x] v)
2 ellam 
a, b e. \ x, v <-> E. x a, b = x, v
3 1, 2 ax_mp 
(E. x a, b = x, v <-> b = N[a / x] v) -> (a, b e. \ x, v <-> b = N[a / x] v)
4 bitr 
(E. x a, b = x, v <-> E. x (x = a /\ b = v)) -> (E. x (x = a /\ b = v) <-> b = N[a / x] v) -> (E. x a, b = x, v <-> b = N[a / x] v)
5 bitr 
(a, b = x, v <-> a = x /\ b = v) -> (a = x /\ b = v <-> x = a /\ b = v) -> (a, b = x, v <-> x = a /\ b = v)
6 prth 
a, b = x, v <-> a = x /\ b = v
7 5, 6 ax_mp 
(a = x /\ b = v <-> x = a /\ b = v) -> (a, b = x, v <-> x = a /\ b = v)
8 eqcomb 
a = x <-> x = a
9 8 aneq1i 
a = x /\ b = v <-> x = a /\ b = v
10 7, 9 ax_mp 
a, b = x, v <-> x = a /\ b = v
11 10 exeqi 
E. x a, b = x, v <-> E. x (x = a /\ b = v)
12 4, 11 ax_mp 
(E. x (x = a /\ b = v) <-> b = N[a / x] v) -> (E. x a, b = x, v <-> b = N[a / x] v)
13 bitr3 
([a / x] b = v <-> E. x (x = a /\ b = v)) -> ([a / x] b = v <-> b = N[a / x] v) -> (E. x (x = a /\ b = v) <-> b = N[a / x] v)
14 dfsb3 
[a / x] b = v <-> E. x (x = a /\ b = v)
15 13, 14 ax_mp 
([a / x] b = v <-> b = N[a / x] v) -> (E. x (x = a /\ b = v) <-> b = N[a / x] v)
16 nfnv 
FN/ x b
17 nfsbn1 
FN/ x N[a / x] v
18 16, 17 nf_eq 
F/ x b = N[a / x] v
19 sbnq 
x = a -> v = N[a / x] v
20 19 eqeq2d 
x = a -> (b = v <-> b = N[a / x] v)
21 18, 20 sbeh 
[a / x] b = v <-> b = N[a / x] v
22 15, 21 ax_mp 
E. x (x = a /\ b = v) <-> b = N[a / x] v
23 12, 22 ax_mp 
E. x a, b = x, v <-> b = N[a / x] v
24 3, 23 ax_mp 
a, b e. \ x, v <-> b = N[a / x] v

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)