Theorem eqtheabd ≪ | index | src | ≫ 

theorem eqtheabd (G: wff) (a: nat) {x: nat} (p: wff x):
  $ G -> (p <-> x = a) $ >
  $ G -> the {x | p} = a $;
StepHypRefExpression
1 elab 
y e. {x | p} <-> [y / x] p
2 hyp h 
G -> (p <-> x = a)
3 2 iald 
G -> A. x (p <-> x = a)
4 nfsb1 
F/ x [y / x] p
5 nfv 
F/ x y = a
6 4, 5 nfbi 
F/ x [y / x] p <-> y = a
7 sbq 
x = y -> (p <-> [y / x] p)
8 eqeq1 
x = y -> (x = a <-> y = a)
9 7, 8 bieqd 
x = y -> (p <-> x = a <-> ([y / x] p <-> y = a))
10 6, 9 ealeh 
A. x (p <-> x = a) -> ([y / x] p <-> y = a)
11 3, 10 rsyl 
G -> ([y / x] p <-> y = a)
12 1, 11 syl5bb 
G -> (y e. {x | p} <-> y = a)
13 12 eqthed 
G -> the {x | p} = 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)