Step |
Hyp |
Ref |
Expression |
1 |
|
brpprod3.1 |
|- X e. _V |
2 |
|
brpprod3.2 |
|- Y e. _V |
3 |
|
brpprod3.3 |
|- Z e. _V |
4 |
|
pprodcnveq |
|- pprod ( R , S ) = `' pprod ( `' R , `' S ) |
5 |
4
|
breqi |
|- ( X pprod ( R , S ) <. Y , Z >. <-> X `' pprod ( `' R , `' S ) <. Y , Z >. ) |
6 |
|
opex |
|- <. Y , Z >. e. _V |
7 |
1 6
|
brcnv |
|- ( X `' pprod ( `' R , `' S ) <. Y , Z >. <-> <. Y , Z >. pprod ( `' R , `' S ) X ) |
8 |
2 3 1
|
brpprod3a |
|- ( <. Y , Z >. pprod ( `' R , `' S ) X <-> E. z E. w ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) ) |
9 |
7 8
|
bitri |
|- ( X `' pprod ( `' R , `' S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) ) |
10 |
|
biid |
|- ( X = <. z , w >. <-> X = <. z , w >. ) |
11 |
|
vex |
|- z e. _V |
12 |
2 11
|
brcnv |
|- ( Y `' R z <-> z R Y ) |
13 |
|
vex |
|- w e. _V |
14 |
3 13
|
brcnv |
|- ( Z `' S w <-> w S Z ) |
15 |
10 12 14
|
3anbi123i |
|- ( ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) <-> ( X = <. z , w >. /\ z R Y /\ w S Z ) ) |
16 |
15
|
2exbii |
|- ( E. z E. w ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) ) |
17 |
9 16
|
bitri |
|- ( X `' pprod ( `' R , `' S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) ) |
18 |
5 17
|
bitri |
|- ( X pprod ( R , S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) ) |