Theorem writeEq ≪ | index | src | ≫ 

pub theorem writeEq (F: set) (a b: nat): $ write F a b @ a = b $;
StepHypRefExpression
1 bitr 
(a, y e. write F a b <-> ifp (a = a) (y = b) (a, y e. F)) -> (ifp (a = a) (y = b) (a, y e. F) <-> y = b) -> (a, y e. write F a b <-> y = b)
2 elwrite 
a, y e. write F a b <-> ifp (a = a) (y = b) (a, y e. F)
3 1, 2 ax_mp 
(ifp (a = a) (y = b) (a, y e. F) <-> y = b) -> (a, y e. write F a b <-> y = b)
4 ifppos 
a = a -> (ifp (a = a) (y = b) (a, y e. F) <-> y = b)
5 eqid 
a = a
6 4, 5 ax_mp 
ifp (a = a) (y = b) (a, y e. F) <-> y = b
7 3, 6 ax_mp 
a, y e. write F a b <-> y = b
8 7 a1i 
T. -> (a, y e. write F a b <-> y = b)
9 8 eqtheabd 
T. -> the {y | a, y e. write F a b} = b
10 9 conv app 
T. -> write F a b @ a = b
11 10 trud 
write F a b @ a = b

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)