Theorem subSS | index | src |

theorem subSS (a b: nat): $ suc a - suc b = a - b $;
StepHypRefExpression
1 eqtr3
a + 1 - (b + 1) = suc a - suc b -> a + 1 - (b + 1) = a - b -> suc a - suc b = a - b
2 subeq
a + 1 = suc a -> b + 1 = suc b -> a + 1 - (b + 1) = suc a - suc b
3 add12
a + 1 = suc a
4 2, 3 ax_mp
b + 1 = suc b -> a + 1 - (b + 1) = suc a - suc b
5 add12
b + 1 = suc b
6 4, 5 ax_mp
a + 1 - (b + 1) = suc a - suc b
7 1, 6 ax_mp
a + 1 - (b + 1) = a - b -> suc a - suc b = a - b
8 pnpcan2
a + 1 - (b + 1) = a - b
9 7, 8 ax_mp
suc a - suc b = a - b

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 (peano2, peano5, addeq, add0, addS)