Theorem add4 ≪ | index | src | ≫

theorem add4 (a b c d: nat): $ a + b + (c + d) = a + c + (b + d) $;

Step	Hyp	Ref	Expression
1		eqtr3	a + b + c + d = a + b + (c + d) -> a + b + c + d = a + c + (b + d) -> a + b + (c + d) = a + c + (b + d)
2		addass	a + b + c + d = a + b + (c + d)
3	1, 2	ax_mp	a + b + c + d = a + c + (b + d) -> a + b + (c + d) = a + c + (b + d)
4		eqtr	a + b + c + d = a + c + b + d -> a + c + b + d = a + c + (b + d) -> a + b + c + d = a + c + (b + d)
5		addeq1	a + b + c = a + c + b -> a + b + c + d = a + c + b + d
6		addrass	a + b + c = a + c + b
7	5, 6	ax_mp	a + b + c + d = a + c + b + d
8	4, 7	ax_mp	a + c + b + d = a + c + (b + d) -> a + b + c + d = a + c + (b + d)
9		addass	a + c + b + d = a + c + (b + d)
10	8, 9	ax_mp	a + b + c + d = a + c + (b + d)
11	3, 10	ax_mp	a + b + (c + d) = a + c + (b + 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_peano (peano2, peano5, addeq, add0, addS)