Theorem pi222pr | index | src |

theorem pi222pr (a b c d: nat): $ pi222 (a, b, c, d) = d $;
StepHypRefExpression
1 eqtr
pi222 (a, b, c, d) = snd (c, d) -> snd (c, d) = d -> pi222 (a, b, c, d) = d
2 sndeq
pi22 (a, b, c, d) = c, d -> snd (pi22 (a, b, c, d)) = snd (c, d)
3 2 conv pi222
pi22 (a, b, c, d) = c, d -> pi222 (a, b, c, d) = snd (c, d)
4 pi22pr
pi22 (a, b, c, d) = c, d
5 3, 4 ax_mp
pi222 (a, b, c, d) = snd (c, d)
6 1, 5 ax_mp
snd (c, d) = d -> pi222 (a, b, c, d) = d
7 sndpr
snd (c, d) = d
8 6, 7 ax_mp
pi222 (a, b, c, d) = d

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)