Theorem eqzsub2 ≪ | index | src | ≫

theorem eqzsub2 (a b c: nat): $ a -Z b = c <-> b +Z c = a $;

Step	Hyp	Ref	Expression
1		bitr	(a -Z b = c <-> c +Z b = a) -> (c +Z b = a <-> b +Z c = a) -> (a -Z b = c <-> b +Z c = a)
2		eqzsub	a -Z b = c <-> c +Z b = a
3	1, 2	ax_mp	(c +Z b = a <-> b +Z c = a) -> (a -Z b = c <-> b +Z c = a)
4		eqeq1	c +Z b = b +Z c -> (c +Z b = a <-> b +Z c = a)
5		zaddcom	c +Z b = b +Z c
6	4, 5	ax_mp	c +Z b = a <-> b +Z c = a
7	3, 6	ax_mp	a -Z b = c <-> b +Z c = 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, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)