Step |
Hyp |
Ref |
Expression |
1 |
|
nfna1 |
|- F/ x -. A. x x = t |
2 |
|
nfeqf2 |
|- ( -. A. x x = t -> F/ x y = t ) |
3 |
1 2
|
nfan1 |
|- F/ x ( -. A. x x = t /\ y = t ) |
4 |
|
equequ2 |
|- ( y = t -> ( x = y <-> x = t ) ) |
5 |
4
|
imbi1d |
|- ( y = t -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) ) |
6 |
5
|
adantl |
|- ( ( -. A. x x = t /\ y = t ) -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) ) |
7 |
3 6
|
albid |
|- ( ( -. A. x x = t /\ y = t ) -> ( A. x ( x = y -> ph ) <-> A. x ( x = t -> ph ) ) ) |
8 |
7
|
pm5.74da |
|- ( -. A. x x = t -> ( ( y = t -> A. x ( x = y -> ph ) ) <-> ( y = t -> A. x ( x = t -> ph ) ) ) ) |
9 |
8
|
albidv |
|- ( -. A. x x = t -> ( A. y ( y = t -> A. x ( x = y -> ph ) ) <-> A. y ( y = t -> A. x ( x = t -> ph ) ) ) ) |
10 |
|
df-sb |
|- ( [ t / x ] ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) ) |
11 |
|
ax6ev |
|- E. y y = t |
12 |
11
|
a1bi |
|- ( A. x ( x = t -> ph ) <-> ( E. y y = t -> A. x ( x = t -> ph ) ) ) |
13 |
|
19.23v |
|- ( A. y ( y = t -> A. x ( x = t -> ph ) ) <-> ( E. y y = t -> A. x ( x = t -> ph ) ) ) |
14 |
12 13
|
bitr4i |
|- ( A. x ( x = t -> ph ) <-> A. y ( y = t -> A. x ( x = t -> ph ) ) ) |
15 |
9 10 14
|
3bitr4g |
|- ( -. A. x x = t -> ( [ t / x ] ph <-> A. x ( x = t -> ph ) ) ) |