Step |
Hyp |
Ref |
Expression |
1 |
|
dfclel |
|- ( x e. { y | ph } <-> E. z ( z = x /\ z e. { y | ph } ) ) |
2 |
|
wl-clabv |
|- ( z e. { y | ph } <-> [ z / y ] ph ) |
3 |
|
sbequ |
|- ( z = x -> ( [ z / y ] ph <-> [ x / y ] ph ) ) |
4 |
2 3
|
syl5bb |
|- ( z = x -> ( z e. { y | ph } <-> [ x / y ] ph ) ) |
5 |
4
|
pm5.32i |
|- ( ( z = x /\ z e. { y | ph } ) <-> ( z = x /\ [ x / y ] ph ) ) |
6 |
5
|
exbii |
|- ( E. z ( z = x /\ z e. { y | ph } ) <-> E. z ( z = x /\ [ x / y ] ph ) ) |
7 |
|
19.41v |
|- ( E. z ( z = x /\ [ x / y ] ph ) <-> ( E. z z = x /\ [ x / y ] ph ) ) |
8 |
|
ax6ev |
|- E. z z = x |
9 |
8
|
biantrur |
|- ( [ x / y ] ph <-> ( E. z z = x /\ [ x / y ] ph ) ) |
10 |
7 9
|
bitr4i |
|- ( E. z ( z = x /\ [ x / y ] ph ) <-> [ x / y ] ph ) |
11 |
1 6 10
|
3bitri |
|- ( x e. { y | ph } <-> [ x / y ] ph ) |