Theorem allsnoc ≪ | index | src | ≫

theorem allsnoc (A: set) (a b: nat): $ all A (a |> b) <-> all A a /\ b e. A $;

Step	Hyp	Ref	Expression
1		bitr	(all A (a \|> b) <-> all A a /\ all A (b : 0)) -> (all A a /\ all A (b : 0) <-> all A a /\ b e. A) -> (all A (a \|> b) <-> all A a /\ b e. A)
2		allappend	all A (a ++ b : 0) <-> all A a /\ all A (b : 0)
3	2	conv snoc	all A (a \|> b) <-> all A a /\ all A (b : 0)
4	1, 3	ax_mp	(all A a /\ all A (b : 0) <-> all A a /\ b e. A) -> (all A (a \|> b) <-> all A a /\ b e. A)
5		all1	all A (b : 0) <-> b e. A
6	5	aneq2i	all A a /\ all A (b : 0) <-> all A a /\ b e. A
7	4, 6	ax_mp	all A (a \|> b) <-> all A a /\ b e. 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)