Theorem eqmeqm23d ≪ | index | src | ≫

theorem eqmeqm23d (G: wff) (a b c d n: nat):
  $ G -> mod(n): a = b $ >
  $ G -> mod(n): c = d $ >
  $ G -> (mod(n): a = c <-> mod(n): b = d) $;

Step	Hyp	Ref	Expression
1		eqmtr	mod(n): b = a -> mod(n): a = d -> mod(n): b = d
2		eqmcom	mod(n): a = b -> mod(n): b = a
3		hyp h1	G -> mod(n): a = b
4	3	anwl	G /\ mod(n): a = c -> mod(n): a = b
5	2, 4	syl	G /\ mod(n): a = c -> mod(n): b = a
6		eqmtr	mod(n): a = c -> mod(n): c = d -> mod(n): a = d
7		anr	G /\ mod(n): a = c -> mod(n): a = c
8		hyp h2	G -> mod(n): c = d
9	8	anwl	G /\ mod(n): a = c -> mod(n): c = d
10	6, 7, 9	sylc	G /\ mod(n): a = c -> mod(n): a = d
11	1, 5, 10	sylc	G /\ mod(n): a = c -> mod(n): b = d
12		eqmtr	mod(n): a = b -> mod(n): b = c -> mod(n): a = c
13	3	anwl	G /\ mod(n): b = d -> mod(n): a = b
14		eqmtr	mod(n): b = d -> mod(n): d = c -> mod(n): b = c
15		anr	G /\ mod(n): b = d -> mod(n): b = d
16		eqmcom	mod(n): c = d -> mod(n): d = c
17	8	anwl	G /\ mod(n): b = d -> mod(n): c = d
18	16, 17	syl	G /\ mod(n): b = d -> mod(n): d = c
19	14, 15, 18	sylc	G /\ mod(n): b = d -> mod(n): b = c
20	12, 13, 19	sylc	G /\ mod(n): b = d -> mod(n): a = c
21	11, 20	ibida	G -> (mod(n): a = c <-> mod(n): b = d)

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)