Metamath Proof Explorer

Theorem cbvoprab13davw

Description: Change the first and third bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025)

Ref Expression
Hypothesis cbvoprab13davw.1 
|- ( ( ( ph /\ x = w ) /\ z = v ) -> ( ps <-> ch ) )
Assertion cbvoprab13davw 
|- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. w , y >. , v >. | ch } )

Proof

Step Hyp Ref Expression
1 cbvoprab13davw.1 
 |-  ( ( ( ph /\ x = w ) /\ z = v ) -> ( ps <-> ch ) )
2 simplr 
 |-  ( ( ( ph /\ x = w ) /\ z = v ) -> x = w )
3 eqidd 
 |-  ( ( ( ph /\ x = w ) /\ z = v ) -> y = y )
4 2 3 opeq12d 
 |-  ( ( ( ph /\ x = w ) /\ z = v ) -> <. x , y >. = <. w , y >. )
5 simpr 
 |-  ( ( ( ph /\ x = w ) /\ z = v ) -> z = v )
6 4 5 opeq12d 
 |-  ( ( ( ph /\ x = w ) /\ z = v ) -> <. <. x , y >. , z >. = <. <. w , y >. , v >. )
7 6 eqeq2d 
 |-  ( ( ( ph /\ x = w ) /\ z = v ) -> ( t = <. <. x , y >. , z >. <-> t = <. <. w , y >. , v >. ) )
8 7 1 anbi12d 
 |-  ( ( ( ph /\ x = w ) /\ z = v ) -> ( ( t = <. <. x , y >. , z >. /\ ps ) <-> ( t = <. <. w , y >. , v >. /\ ch ) ) )
9 8 cbvexdvaw 
 |-  ( ( ph /\ x = w ) -> ( E. z ( t = <. <. x , y >. , z >. /\ ps ) <-> E. v ( t = <. <. w , y >. , v >. /\ ch ) ) )
10 9 exbidv 
 |-  ( ( ph /\ x = w ) -> ( E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) <-> E. y E. v ( t = <. <. w , y >. , v >. /\ ch ) ) )
11 10 cbvexdvaw 
 |-  ( ph -> ( E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) <-> E. w E. y E. v ( t = <. <. w , y >. , v >. /\ ch ) ) )
12 11 abbidv 
 |-  ( ph -> { t | E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) } = { t | E. w E. y E. v ( t = <. <. w , y >. , v >. /\ ch ) } )
13 df-oprab 
 |-  { <. <. x , y >. , z >. | ps } = { t | E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) }
14 df-oprab 
 |-  { <. <. w , y >. , v >. | ch } = { t | E. w E. y E. v ( t = <. <. w , y >. , v >. /\ ch ) }
15 12 13 14 3eqtr4g 
 |-  ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. w , y >. , v >. | ch } )