Theorem ltpr1 ≪ | index | src | ≫

theorem ltpr1 (a b c: nat): $ a < b <-> a, c < b, c $;

Step	Hyp	Ref	Expression
1		ltnle	a < b <-> ~b <= a
2		ltnle	a, c < b, c <-> ~b, c <= a, c
3		noteq	(b <= a <-> b, c <= a, c) -> (~b <= a <-> ~b, c <= a, c)
4		lepr1	b <= a <-> b, c <= a, c
5	3, 4	ax_mp	~b <= a <-> ~b, c <= a, c
6	1, 2, 5	bitr4gi	a < b <-> a, c < b, 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)