Theorem pi12pr | index | src |

theorem pi12pr (a b c: nat): $ pi12 ((a, b), c) = b $;
StepHypRefExpression
1 eqtr
pi12 ((a, b), c) = snd (a, b) -> snd (a, b) = b -> pi12 ((a, b), c) = b
2 sndeq
fst ((a, b), c) = a, b -> snd (fst ((a, b), c)) = snd (a, b)
3 2 conv pi12
fst ((a, b), c) = a, b -> pi12 ((a, b), c) = snd (a, b)
4 fstpr
fst ((a, b), c) = a, b
5 3, 4 ax_mp
pi12 ((a, b), c) = snd (a, b)
6 1, 5 ax_mp
snd (a, b) = b -> pi12 ((a, b), c) = b
7 sndpr
snd (a, b) = b
8 6, 7 ax_mp
pi12 ((a, b), c) = 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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)