Theorem zeqmeqm23d ≪ | index | src | ≫

theorem zeqmeqm23d (G: wff) (a b c d n: nat):
  $ G -> modZ(n): a = b $ >
  $ G -> modZ(n): c = d $ >
  $ G -> (modZ(n): a = c <-> modZ(n): b = d) $;

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

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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)