Step |
Hyp |
Ref |
Expression |
1 |
|
cbvopab1g.1 |
|- F/ z ph |
2 |
|
cbvopab1g.2 |
|- F/ x ps |
3 |
|
cbvopab1g.3 |
|- ( x = z -> ( ph <-> ps ) ) |
4 |
|
nfv |
|- F/ v E. y ( w = <. x , y >. /\ ph ) |
5 |
|
nfv |
|- F/ x w = <. v , y >. |
6 |
|
nfs1v |
|- F/ x [ v / x ] ph |
7 |
5 6
|
nfan |
|- F/ x ( w = <. v , y >. /\ [ v / x ] ph ) |
8 |
7
|
nfex |
|- F/ x E. y ( w = <. v , y >. /\ [ v / x ] ph ) |
9 |
|
opeq1 |
|- ( x = v -> <. x , y >. = <. v , y >. ) |
10 |
9
|
eqeq2d |
|- ( x = v -> ( w = <. x , y >. <-> w = <. v , y >. ) ) |
11 |
|
sbequ12 |
|- ( x = v -> ( ph <-> [ v / x ] ph ) ) |
12 |
10 11
|
anbi12d |
|- ( x = v -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. v , y >. /\ [ v / x ] ph ) ) ) |
13 |
12
|
exbidv |
|- ( x = v -> ( E. y ( w = <. x , y >. /\ ph ) <-> E. y ( w = <. v , y >. /\ [ v / x ] ph ) ) ) |
14 |
4 8 13
|
cbvexv1 |
|- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. v E. y ( w = <. v , y >. /\ [ v / x ] ph ) ) |
15 |
|
nfv |
|- F/ z w = <. v , y >. |
16 |
1
|
nfsb |
|- F/ z [ v / x ] ph |
17 |
15 16
|
nfan |
|- F/ z ( w = <. v , y >. /\ [ v / x ] ph ) |
18 |
17
|
nfex |
|- F/ z E. y ( w = <. v , y >. /\ [ v / x ] ph ) |
19 |
|
nfv |
|- F/ v E. y ( w = <. z , y >. /\ ps ) |
20 |
|
opeq1 |
|- ( v = z -> <. v , y >. = <. z , y >. ) |
21 |
20
|
eqeq2d |
|- ( v = z -> ( w = <. v , y >. <-> w = <. z , y >. ) ) |
22 |
|
sbequ |
|- ( v = z -> ( [ v / x ] ph <-> [ z / x ] ph ) ) |
23 |
2 3
|
sbie |
|- ( [ z / x ] ph <-> ps ) |
24 |
22 23
|
bitrdi |
|- ( v = z -> ( [ v / x ] ph <-> ps ) ) |
25 |
21 24
|
anbi12d |
|- ( v = z -> ( ( w = <. v , y >. /\ [ v / x ] ph ) <-> ( w = <. z , y >. /\ ps ) ) ) |
26 |
25
|
exbidv |
|- ( v = z -> ( E. y ( w = <. v , y >. /\ [ v / x ] ph ) <-> E. y ( w = <. z , y >. /\ ps ) ) ) |
27 |
18 19 26
|
cbvexv1 |
|- ( E. v E. y ( w = <. v , y >. /\ [ v / x ] ph ) <-> E. z E. y ( w = <. z , y >. /\ ps ) ) |
28 |
14 27
|
bitri |
|- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ ps ) ) |
29 |
28
|
abbii |
|- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ ps ) } |
30 |
|
df-opab |
|- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) } |
31 |
|
df-opab |
|- { <. z , y >. | ps } = { w | E. z E. y ( w = <. z , y >. /\ ps ) } |
32 |
29 30 31
|
3eqtr4i |
|- { <. x , y >. | ph } = { <. z , y >. | ps } |