Theorem pi221pr | index | src |

theorem pi221pr (a b c d: nat): $ pi221 (a, b, c, d) = c $;
StepHypRefExpression
1 eqtr
pi221 (a, b, c, d) = fst (c, d) -> fst (c, d) = c -> pi221 (a, b, c, d) = c
2 fsteq
pi22 (a, b, c, d) = c, d -> fst (pi22 (a, b, c, d)) = fst (c, d)
3 2 conv pi221
pi22 (a, b, c, d) = c, d -> pi221 (a, b, c, d) = fst (c, d)
4 pi22pr
pi22 (a, b, c, d) = c, d
5 3, 4 ax_mp
pi221 (a, b, c, d) = fst (c, d)
6 1, 5 ax_mp
fst (c, d) = c -> pi221 (a, b, c, d) = c
7 fstpr
fst (c, d) = c
8 6, 7 ax_mp
pi221 (a, b, c, d) = c

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)