Theorem eqthe0abd ≪ | index | src | ≫ 

theorem eqthe0abd (G: wff) {x: nat} (p: wff x):
  $ G -> p -> x = 0 $ >
  $ G -> the {x | p} = 0 $;
StepHypRefExpression
1 elab 
y e. {x | p} <-> [y / x] p
2 hyp h 
G -> p -> x = 0
3 2 iald 
G -> A. x (p -> x = 0)
4 nfsb1 
F/ x [y / x] p
5 nfv 
F/ x y = 0
6 4, 5 nfim 
F/ x [y / x] p -> y = 0
7 sbq 
x = y -> (p <-> [y / x] p)
8 eqeq1 
x = y -> (x = 0 <-> y = 0)
9 7, 8 imeqd 
x = y -> (p -> x = 0 <-> [y / x] p -> y = 0)
10 6, 9 ealeh 
A. x (p -> x = 0) -> [y / x] p -> y = 0
11 3, 10 rsyl 
G -> [y / x] p -> y = 0
12 1, 11 syl5bi 
G -> y e. {x | p} -> y = 0
13 12 eqthe0d 
G -> the {x | p} = 0

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)