Metamath Proof Explorer

Theorem cbvoprab2davw

Description: Change the second bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025)

Ref Expression
Hypothesis cbvoprab2davw.1 
|- ( ( ph /\ y = w ) -> ( ps <-> ch ) )
Assertion cbvoprab2davw 
|- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , w >. , z >. | ch } )

Proof

Step Hyp Ref Expression
1 cbvoprab2davw.1 
 |-  ( ( ph /\ y = w ) -> ( ps <-> ch ) )
2 opeq2 
 |-  ( y = w -> <. x , y >. = <. x , w >. )
3 2 adantl 
 |-  ( ( ph /\ y = w ) -> <. x , y >. = <. x , w >. )
4 3 opeq1d 
 |-  ( ( ph /\ y = w ) -> <. <. x , y >. , z >. = <. <. x , w >. , z >. )
5 4 eqeq2d 
 |-  ( ( ph /\ y = w ) -> ( t = <. <. x , y >. , z >. <-> t = <. <. x , w >. , z >. ) )
6 5 1 anbi12d 
 |-  ( ( ph /\ y = w ) -> ( ( t = <. <. x , y >. , z >. /\ ps ) <-> ( t = <. <. x , w >. , z >. /\ ch ) ) )
7 6 exbidv 
 |-  ( ( ph /\ y = w ) -> ( E. z ( t = <. <. x , y >. , z >. /\ ps ) <-> E. z ( t = <. <. x , w >. , z >. /\ ch ) ) )
8 7 cbvexdvaw 
 |-  ( ph -> ( E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) <-> E. w E. z ( t = <. <. x , w >. , z >. /\ ch ) ) )
9 8 exbidv 
 |-  ( ph -> ( E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) <-> E. x E. w E. z ( t = <. <. x , w >. , z >. /\ ch ) ) )
10 9 abbidv 
 |-  ( ph -> { t | E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) } = { t | E. x E. w E. z ( t = <. <. x , w >. , z >. /\ ch ) } )
11 df-oprab 
 |-  { <. <. x , y >. , z >. | ps } = { t | E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) }
12 df-oprab 
 |-  { <. <. x , w >. , z >. | ch } = { t | E. x E. w E. z ( t = <. <. x , w >. , z >. /\ ch ) }
13 10 11 12 3eqtr4g 
 |-  ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , w >. , z >. | ch } )