Theorem consinj ≪ | index | src | ≫

theorem consinj (a b c d: nat): $ a : c = b : d <-> a = b /\ c = d $;

Step	Hyp	Ref	Expression
1		bitr	(a : c = b : d <-> a, c = b, d) -> (a, c = b, d <-> a = b /\ c = d) -> (a : c = b : d <-> a = b /\ c = d)
2		peano2	suc (a, c) = suc (b, d) <-> a, c = b, d
3	2	conv cons	a : c = b : d <-> a, c = b, d
4	1, 3	ax_mp	(a, c = b, d <-> a = b /\ c = d) -> (a : c = b : d <-> a = b /\ c = d)
5		prth	a, c = b, d <-> a = b /\ c = d
6	4, 5	ax_mp	a : c = b : d <-> a = b /\ c = 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)