Theorem bool01 ≪ | index | src | ≫ 

theorem bool01 (n: nat): $ bool n <-> n = 0 \/ n = 1 $;
StepHypRefExpression
1 bitr3 
(n <= 1 <-> bool n) -> (n <= 1 <-> n = 0 \/ n = 1) -> (bool n <-> n = 0 \/ n = 1)
2 leltsuc 
n <= 1 <-> n < suc 1
3 2 conv bool, d2 
n <= 1 <-> bool n
4 1, 3 ax_mp 
(n <= 1 <-> n = 0 \/ n = 1) -> (bool n <-> n = 0 \/ n = 1)
5 bitr 
(n <= 1 <-> n < 1 \/ n = 1) -> (n < 1 \/ n = 1 <-> n = 0 \/ n = 1) -> (n <= 1 <-> n = 0 \/ n = 1)
6 leloe 
n <= 1 <-> n < 1 \/ n = 1
7 5, 6 ax_mp 
(n < 1 \/ n = 1 <-> n = 0 \/ n = 1) -> (n <= 1 <-> n = 0 \/ n = 1)
8 lt12 
n < 1 <-> n = 0
9 8 oreq1i 
n < 1 \/ n = 1 <-> n = 0 \/ n = 1
10 7, 9 ax_mp 
n <= 1 <-> n = 0 \/ n = 1
11 4, 10 ax_mp 
bool n <-> n = 0 \/ n = 1

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_peano (peano1, peano2, peano5, addeq, add0, addS)