Theorem subsnex ≪ | index | src | ≫

theorem subsnex (A: set) {a x: nat}:
  $ subsn A <-> E. a A. x (x e. A -> x = a) $;

Step	Hyp	Ref	Expression
1		id	a = the A -> a = the A
2	1	eqeq2d	a = the A -> (x = a <-> x = the A)
3	2	imeq2d	a = the A -> (x e. A -> x = a <-> x e. A -> x = the A)
4	3	aleqd	a = the A -> (A. x (x e. A -> x = a) <-> A. x (x e. A -> x = the A))
5	4	iexe	A. x (x e. A -> x = the A) -> E. a A. x (x e. A -> x = a)
6		eqcom	the A = x -> x = the A
7		subsnthe	subsn A -> x e. A -> the A = x
8	6, 7	syl6	subsn A -> x e. A -> x = the A
9	8	iald	subsn A -> A. x (x e. A -> x = the A)
10	5, 9	syl	subsn A -> E. a A. x (x e. A -> x = a)
11		id	x = a1 -> x = a1
12	11	eleq1d	x = a1 -> (x e. A <-> a1 e. A)
13	11	eqeq1d	x = a1 -> (x = a <-> a1 = a)
14	12, 13	imeqd	x = a1 -> (x e. A -> x = a <-> a1 e. A -> a1 = a)
15	14	eale	A. x (x e. A -> x = a) -> a1 e. A -> a1 = a
16	15	eqsubsnd	A. x (x e. A -> x = a) -> subsn A
17	16	eex	E. a A. x (x e. A -> x = a) -> subsn A
18	10, 17	ibii	subsn A <-> E. a A. x (x e. A -> x = a)

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)