Step |
Hyp |
Ref |
Expression |
1 |
|
nfsbc1v |
|- F/ x [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph |
2 |
1
|
19.41 |
|- ( E. x ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( E. x E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) ) |
3 |
|
sbcopeq1a |
|- ( z = <. x , y >. -> ( [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph <-> ph ) ) |
4 |
3
|
pm5.32i |
|- ( ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( z = <. x , y >. /\ ph ) ) |
5 |
4
|
exbii |
|- ( E. y ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> E. y ( z = <. x , y >. /\ ph ) ) |
6 |
|
nfcv |
|- F/_ y ( 1st ` z ) |
7 |
|
nfsbc1v |
|- F/ y [. ( 2nd ` z ) / y ]. ph |
8 |
6 7
|
nfsbcw |
|- F/ y [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph |
9 |
8
|
19.41 |
|- ( E. y ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) ) |
10 |
5 9
|
bitr3i |
|- ( E. y ( z = <. x , y >. /\ ph ) <-> ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) ) |
11 |
10
|
exbii |
|- ( E. x E. y ( z = <. x , y >. /\ ph ) <-> E. x ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) ) |
12 |
|
elvv |
|- ( z e. ( _V X. _V ) <-> E. x E. y z = <. x , y >. ) |
13 |
12
|
anbi1i |
|- ( ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( E. x E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) ) |
14 |
2 11 13
|
3bitr4i |
|- ( E. x E. y ( z = <. x , y >. /\ ph ) <-> ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) ) |
15 |
14
|
abbii |
|- { z | E. x E. y ( z = <. x , y >. /\ ph ) } = { z | ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) } |
16 |
|
df-opab |
|- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) } |
17 |
|
df-rab |
|- { z e. ( _V X. _V ) | [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph } = { z | ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) } |
18 |
15 16 17
|
3eqtr4i |
|- { <. x , y >. | ph } = { z e. ( _V X. _V ) | [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph } |