Theorem oddS ≪ | index | src | ≫ 

theorem oddS (n: nat): $ odd (suc n) <-> ~odd n $;
StepHypRefExpression
1 bitr 
(odd (suc n) <-> ~2 || suc n) -> (~2 || suc n <-> ~odd n) -> (odd (suc n) <-> ~odd n)
2 odddvd 
odd (suc n) <-> ~2 || suc n
3 1, 2 ax_mp 
(~2 || suc n <-> ~odd n) -> (odd (suc n) <-> ~odd n)
4 noteq 
(2 || suc n <-> odd n) -> (~2 || suc n <-> ~odd n)
5 bitr4 
(2 || suc n <-> ~2 || n) -> (odd n <-> ~2 || n) -> (2 || suc n <-> odd n)
6 d2dvdS 
2 || suc n <-> ~2 || n
7 5, 6 ax_mp 
(odd n <-> ~2 || n) -> (2 || suc n <-> odd n)
8 odddvd 
odd n <-> ~2 || n
9 7, 8 ax_mp 
2 || suc n <-> odd n
10 4, 9 ax_mp 
~2 || suc n <-> ~odd n
11 3, 10 ax_mp 
odd (suc n) <-> ~odd n

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)