Theorem eqb1 ≪ | index | src | ≫ 

theorem eqb1 (n: nat): $ odd n <-> n = b1 (n // 2) $;
StepHypRefExpression
1 b0orb1 
n = b0 (n // 2) \/ n = b1 (n // 2)
2 1 conv or 
~n = b0 (n // 2) -> n = b1 (n // 2)
3 con2 
(n = b0 (n // 2) -> ~odd n) -> odd n -> ~n = b0 (n // 2)
4 b0odd 
~odd (b0 (n // 2))
5 oddeq 
n = b0 (n // 2) -> (odd n <-> odd (b0 (n // 2)))
6 5 noteqd 
n = b0 (n // 2) -> (~odd n <-> ~odd (b0 (n // 2)))
7 4, 6 mpbiri 
n = b0 (n // 2) -> ~odd n
8 3, 7 ax_mp 
odd n -> ~n = b0 (n // 2)
9 2, 8 syl 
odd n -> n = b1 (n // 2)
10 b1odd 
odd (b1 (n // 2))
11 oddeq 
n = b1 (n // 2) -> (odd n <-> odd (b1 (n // 2)))
12 10, 11 mpbiri 
n = b1 (n // 2) -> odd n
13 9, 12 ibii 
odd n <-> n = b1 (n // 2)

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)