Theorem rappsabs ≪ | index | src | ≫

theorem rappsabs (a: nat) {x: nat} (A: set x): $ (S\ x, A) @' a == S[a / x] A $;

Step	Hyp	Ref	Expression
1		bitr	(a1 e. (S\ x, A) @' a <-> a, a1 e. S\ x, A) -> (a, a1 e. S\ x, A <-> a1 e. S[a / x] A) -> (a1 e. (S\ x, A) @' a <-> a1 e. S[a / x] A)
2		elrapp	a1 e. (S\ x, A) @' a <-> a, a1 e. S\ x, A
3	1, 2	ax_mp	(a, a1 e. S\ x, A <-> a1 e. S[a / x] A) -> (a1 e. (S\ x, A) @' a <-> a1 e. S[a / x] A)
4		elsabs	a, a1 e. S\ x, A <-> a1 e. S[a / x] A
5	3, 4	ax_mp	a1 e. (S\ x, A) @' a <-> a1 e. S[a / x] A
6	5	ax_gen	A. a1 (a1 e. (S\ x, A) @' a <-> a1 e. S[a / x] A)
7	6	conv eqs	(S\ x, A) @' a == S[a / x] 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)