Step |
Hyp |
Ref |
Expression |
1 |
|
eqcom |
|- ( <. z , w >. = <. x , y >. <-> <. x , y >. = <. z , w >. ) |
2 |
|
vex |
|- x e. _V |
3 |
|
vex |
|- y e. _V |
4 |
2 3
|
opth |
|- ( <. x , y >. = <. z , w >. <-> ( x = z /\ y = w ) ) |
5 |
1 4
|
bitri |
|- ( <. z , w >. = <. x , y >. <-> ( x = z /\ y = w ) ) |
6 |
5
|
anbi1i |
|- ( ( <. z , w >. = <. x , y >. /\ ph ) <-> ( ( x = z /\ y = w ) /\ ph ) ) |
7 |
6
|
2exbii |
|- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. x E. y ( ( x = z /\ y = w ) /\ ph ) ) |
8 |
|
elopab |
|- ( <. z , w >. e. { <. x , y >. | ph } <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) ) |
9 |
|
2sb5 |
|- ( [ z / x ] [ w / y ] ph <-> E. x E. y ( ( x = z /\ y = w ) /\ ph ) ) |
10 |
7 8 9
|
3bitr4i |
|- ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph ) |