Theorem all1 ≪ | index | src | ≫

theorem all1 (A: set) (a: nat): $ all A (a : 0) <-> a e. A $;

Step	Hyp	Ref	Expression
1		bitr	(all A (a : 0) <-> a e. A /\ all A 0) -> (a e. A /\ all A 0 <-> a e. A) -> (all A (a : 0) <-> a e. A)
2		allS	all A (a : 0) <-> a e. A /\ all A 0
3	1, 2	ax_mp	(a e. A /\ all A 0 <-> a e. A) -> (all A (a : 0) <-> a e. A)
4		bian2	all A 0 -> (a e. A /\ all A 0 <-> a e. A)
5		all0	all A 0
6	4, 5	ax_mp	a e. A /\ all A 0 <-> a e. A
7	3, 6	ax_mp	all A (a : 0) <-> a 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)