Theorem lameq ≪ | index | src | ≫

theorem lameq {x: nat} (a b: nat x): $ A. x a = b -> \ x, a == \ x, b $;

Step	Hyp	Ref	Expression
1		abeqb	A. p (E. x p = x, a <-> E. x p = x, b) <-> {p \| E. x p = x, a} == {p \| E. x p = x, b}
2	1	conv lam	A. p (E. x p = x, a <-> E. x p = x, b) <-> \ x, a == \ x, b
3		exeq	A. x (p = x, a <-> p = x, b) -> (E. x p = x, a <-> E. x p = x, b)
4		preq2	a = b -> x, a = x, b
5	4	eqeq2d	a = b -> (p = x, a <-> p = x, b)
6	5	alimi	A. x a = b -> A. x (p = x, a <-> p = x, b)
7	3, 6	syl	A. x a = b -> (E. x p = x, a <-> E. x p = x, b)
8	7	iald	A. x a = b -> A. p (E. x p = x, a <-> E. x p = x, b)
9	2, 8	sylib	A. x a = b -> \ x, a == \ x, 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)