Theorem cbvlamh ≪ | index | src | ≫

theorem cbvlamh {x y: nat} (a b: nat x y):
  $ FN/ y a $ >
  $ FN/ x b $ >
  $ x = y -> a = b $ >
  $ \ x, a == \ y, b $;

Step	Hyp	Ref	Expression
1		nfnv	FN/ y x
2		hyp h1	FN/ y a
3	1, 2	nfpr	FN/ y x, a
4	3	nfeq2	F/ y p1 = x, a
5		nfnv	FN/ x y
6		hyp h2	FN/ x b
7	5, 6	nfpr	FN/ x y, b
8	7	nfeq2	F/ x p1 = y, b
9		id	x = y -> x = y
10		hyp e	x = y -> a = b
11	9, 10	preqd	x = y -> x, a = y, b
12	11	eqeq2d	x = y -> (p1 = x, a <-> p1 = y, b)
13	4, 8, 12	cbvexh	E. x p1 = x, a <-> E. y p1 = y, b
14		eqeq1	p1 = p2 -> (p1 = y, b <-> p2 = y, b)
15	14	exeqd	p1 = p2 -> (E. y p1 = y, b <-> E. y p2 = y, b)
16	13, 15	syl5bb	p1 = p2 -> (E. x p1 = x, a <-> E. y p2 = y, b)
17	16	cbvab	{p1 \| E. x p1 = x, a} == {p2 \| E. y p2 = y, b}
18	17	conv lam	\ x, a == \ y, 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 (peano2, addeq, muleq)