Theorem sbid ≪ | index | src | ≫ 

theorem sbid {x: nat} (a: wff x): $ [x / x] a <-> a $;
StepHypRefExpression
1 ax_6 
E. y y = x
2 nfal1 
F/ y A. y (y = x -> A. x (x = y -> a))
3 nfv 
F/ y a
4 eal 
A. y (y = x -> A. x (x = y -> a)) -> y = x -> A. x (x = y -> a)
5 eqcom 
y = x -> x = y
6 eal 
A. x (x = y -> a) -> x = y -> a
7 6 com12 
x = y -> A. x (x = y -> a) -> a
8 5, 7 rsyl 
y = x -> A. x (x = y -> a) -> a
9 8 a2i 
(y = x -> A. x (x = y -> a)) -> y = x -> a
10 4, 9 rsyl 
A. y (y = x -> A. x (x = y -> a)) -> y = x -> a
11 2, 3, 10 eexdh 
A. y (y = x -> A. x (x = y -> a)) -> E. y y = x -> a
12 1, 11 mpi 
A. y (y = x -> A. x (x = y -> a)) -> a
13 dfsb2 
[y / x] a <-> A. x (x = y -> a)
14 sbq 
x = y -> (a <-> [y / x] a)
15 14, 5 syl 
y = x -> (a <-> [y / x] a)
16 13, 15 syl6bb 
y = x -> (a <-> A. x (x = y -> a))
17 16 bi1d 
y = x -> a -> A. x (x = y -> a)
18 17 com12 
a -> y = x -> A. x (x = y -> a)
19 18 iald 
a -> A. y (y = x -> A. x (x = y -> a))
20 12, 19 ibii 
A. y (y = x -> A. x (x = y -> a)) <-> a
21 20 conv sb 
[x / x] a <-> 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_12)