Theorem theeqd ≪ | index | src | ≫

theorem theeqd (G: wff) (A B: set): $ G -> A == B $ > $ G -> the A = the B $;

Step	Hyp	Ref	Expression
1		theid	A == {x \| x = y} -> the A = y
2	1	anwr	G /\ A == {x \| x = y} -> the A = y
3		theid	B == {x \| x = y} -> the B = y
4		eqstr3	A == B -> A == {x \| x = y} -> B == {x \| x = y}
5		hyp h	G -> A == B
6	4, 5	syl	G -> A == {x \| x = y} -> B == {x \| x = y}
7	6	imp	G /\ A == {x \| x = y} -> B == {x \| x = y}
8	3, 7	syl	G /\ A == {x \| x = y} -> the B = y
9	2, 8	eqtr4d	G /\ A == {x \| x = y} -> the A = the B
10	9	eexda	G -> E. y A == {x \| x = y} -> the A = the B
11		the0	~E. y A == {x \| x = y} -> the A = 0
12	11	anwr	G /\ ~E. y A == {x \| x = y} -> the A = 0
13		the0	~E. y B == {x \| x = y} -> the B = 0
14	5	eqseq1d	G -> (A == {x \| x = y} <-> B == {x \| x = y})
15	14	exeqd	G -> (E. y A == {x \| x = y} <-> E. y B == {x \| x = y})
16	15	noteqd	G -> (~E. y A == {x \| x = y} <-> ~E. y B == {x \| x = y})
17	16	impbi	G /\ ~E. y A == {x \| x = y} -> ~E. y B == {x \| x = y}
18	13, 17	syl	G /\ ~E. y A == {x \| x = y} -> the B = 0
19	12, 18	eqtr4d	G /\ ~E. y A == {x \| x = y} -> the A = the B
20	19	exp	G -> ~E. y A == {x \| x = y} -> the A = the B
21	10, 20	casesd	G -> the A = the B

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_12), axs_set (ax_8), axs_the (theid, the0)