Theorem rlameq2b ≪ | index | src | ≫

theorem rlameq2b (a: nat) {x: nat} (v w: nat x):
  $ \. x e. a, v = \. x e. a, w <-> A. x (x e. a -> v = w) $;

Step	Hyp	Ref	Expression
1		nfnv	FN/ x a
2	1	nfrlam1	FN/ x \. x e. a, v
3	1	nfrlam1	FN/ x \. x e. a, w
4	2, 3	nf_eq	F/ x \. x e. a, v = \. x e. a, w
5		appneq1	\. x e. a, v = \. x e. a, w -> (\. x e. a, v) @ x = (\. x e. a, w) @ x
6		apprlam	x e. a -> (\. x e. a, v) @ x = v
7		apprlam	x e. a -> (\. x e. a, w) @ x = w
8	6, 7	eqeqd	x e. a -> ((\. x e. a, v) @ x = (\. x e. a, w) @ x <-> v = w)
9	8	bi1d	x e. a -> (\. x e. a, v) @ x = (\. x e. a, w) @ x -> v = w
10	9	com12	(\. x e. a, v) @ x = (\. x e. a, w) @ x -> x e. a -> v = w
11	5, 10	rsyl	\. x e. a, v = \. x e. a, w -> x e. a -> v = w
12	4, 11	ialdh	\. x e. a, v = \. x e. a, w -> A. x (x e. a -> v = w)
13		rlameq2a	A. x (x e. a -> v = w) -> \. x e. a, v = \. x e. a, w
14	12, 13	ibii	\. x e. a, v = \. x e. a, w <-> A. x (x e. a -> v = 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)