Theorem eqznegcom | index | src |

theorem eqznegcom (a b: nat): $ a = -uZ b <-> b = -uZ a $;
StepHypRefExpression
1 bitr
(a = -uZ b <-> -uZ b = a) -> (-uZ b = a <-> b = -uZ a) -> (a = -uZ b <-> b = -uZ a)
2 eqcomb
a = -uZ b <-> -uZ b = a
3 1, 2 ax_mp
(-uZ b = a <-> b = -uZ a) -> (a = -uZ b <-> b = -uZ a)
4 bitr
(-uZ b = a <-> -uZ a = b) -> (-uZ a = b <-> b = -uZ a) -> (-uZ b = a <-> b = -uZ a)
5 znegeqcom
-uZ b = a <-> -uZ a = b
6 4, 5 ax_mp
(-uZ a = b <-> b = -uZ a) -> (-uZ b = a <-> b = -uZ a)
7 eqcomb
-uZ a = b <-> b = -uZ a
8 6, 7 ax_mp
-uZ b = a <-> b = -uZ a
9 3, 8 ax_mp
a = -uZ b <-> b = -uZ a

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)