Theorem sbneht ≪ | index | src | ≫ 

theorem sbneht {x: nat} (a: nat) (b c: nat x):
  $ FN/ x c $ >
  $ A. x (x = a -> b = c) -> N[a / x] b = c $;
StepHypRefExpression
1 hyp h 
FN/ x c
2 1 nfeq2 
F/ x y = c
3 2 sbeht 
A. x (x = a -> (y = b <-> y = c)) -> ([a / x] y = b <-> y = c)
4 imim2 
(b = c -> (y = b <-> y = c)) -> (x = a -> b = c) -> x = a -> (y = b <-> y = c)
5 eqeq2 
b = c -> (y = b <-> y = c)
6 4, 5 ax_mp 
(x = a -> b = c) -> x = a -> (y = b <-> y = c)
7 6 alimi 
A. x (x = a -> b = c) -> A. x (x = a -> (y = b <-> y = c))
8 3, 7 syl 
A. x (x = a -> b = c) -> ([a / x] y = b <-> y = c)
9 8 eqtheabd 
A. x (x = a -> b = c) -> the {y | [a / x] y = b} = c
10 9 conv sbn 
A. x (x = a -> b = c) -> N[a / x] 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)