Theorem rlameq2a ≪ | index | src | ≫ 

theorem rlameq2a (a: nat) {x: nat} (v w: nat x):
  $ A. x (x e. a -> v = w) -> \. x e. a, v = \. x e. a, w $;
StepHypRefExpression
1 axext 
\. x e. a, v == \. x e. a, w -> \. x e. a, v = \. x e. a, w
2 elrlam 
y e. \. x e. a, v <-> E. x (x e. a /\ y = x, v)
3 elrlam 
y e. \. x e. a, w <-> E. x (x e. a /\ y = x, w)
4 exeq 
A. x (x e. a /\ y = x, v <-> x e. a /\ y = x, w) -> (E. x (x e. a /\ y = x, v) <-> E. x (x e. a /\ y = x, w))
5 aneq2a 
(x e. a -> (y = x, v <-> y = x, w)) -> (x e. a /\ y = x, v <-> x e. a /\ y = x, w)
6 preq2 
v = w -> x, v = x, w
7 6 eqeq2d 
v = w -> (y = x, v <-> y = x, w)
8 7 imim2i 
(x e. a -> v = w) -> x e. a -> (y = x, v <-> y = x, w)
9 5, 8 syl 
(x e. a -> v = w) -> (x e. a /\ y = x, v <-> x e. a /\ y = x, w)
10 9 alimi 
A. x (x e. a -> v = w) -> A. x (x e. a /\ y = x, v <-> x e. a /\ y = x, w)
11 4, 10 syl 
A. x (x e. a -> v = w) -> (E. x (x e. a /\ y = x, v) <-> E. x (x e. a /\ y = x, w))
12 2, 3, 11 bitr4g 
A. x (x e. a -> v = w) -> (y e. \. x e. a, v <-> y e. \. x e. a, w)
13 12 iald 
A. x (x e. a -> v = w) -> A. y (y e. \. x e. a, v <-> y e. \. x e. a, w)
14 13 conv eqs 
A. x (x e. a -> v = w) -> \. x e. a, v == \. x e. a, w
15 1, 14 syl 
A. x (x e. a -> v = w) -> \. x e. a, v = \. x e. a, w

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)