Metamath Proof Explorer

Theorem cbvoprab123vw

Description: Change all bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025)

Ref Expression
Hypothesis cbvoprab123vw.1 
|- ( ( ( x = w /\ y = u ) /\ z = v ) -> ( ps <-> ch ) )
Assertion cbvoprab123vw 
|- { <. <. x , y >. , z >. | ps } = { <. <. w , u >. , v >. | ch }

Proof

Step Hyp Ref Expression
1 cbvoprab123vw.1 
 |-  ( ( ( x = w /\ y = u ) /\ z = v ) -> ( ps <-> ch ) )
2 simpll 
 |-  ( ( ( x = w /\ y = u ) /\ z = v ) -> x = w )
3 simplr 
 |-  ( ( ( x = w /\ y = u ) /\ z = v ) -> y = u )
4 2 3 opeq12d 
 |-  ( ( ( x = w /\ y = u ) /\ z = v ) -> <. x , y >. = <. w , u >. )
5 simpr 
 |-  ( ( ( x = w /\ y = u ) /\ z = v ) -> z = v )
6 4 5 opeq12d 
 |-  ( ( ( x = w /\ y = u ) /\ z = v ) -> <. <. x , y >. , z >. = <. <. w , u >. , v >. )
7 6 eqeq2d 
 |-  ( ( ( x = w /\ y = u ) /\ z = v ) -> ( t = <. <. x , y >. , z >. <-> t = <. <. w , u >. , v >. ) )
8 7 1 anbi12d 
 |-  ( ( ( x = w /\ y = u ) /\ z = v ) -> ( ( t = <. <. x , y >. , z >. /\ ps ) <-> ( t = <. <. w , u >. , v >. /\ ch ) ) )
9 8 cbvexdvaw 
 |-  ( ( x = w /\ y = u ) -> ( E. z ( t = <. <. x , y >. , z >. /\ ps ) <-> E. v ( t = <. <. w , u >. , v >. /\ ch ) ) )
10 9 cbvex2vw 
 |-  ( E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) <-> E. w E. u E. v ( t = <. <. w , u >. , v >. /\ ch ) )
11 10 abbii 
 |-  { t | E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) } = { t | E. w E. u E. v ( t = <. <. w , u >. , v >. /\ ch ) }
12 df-oprab 
 |-  { <. <. x , y >. , z >. | ps } = { t | E. x E. y E. z ( t = <. <. x , y >. , z >. /\ ps ) }
13 df-oprab 
 |-  { <. <. w , u >. , v >. | ch } = { t | E. w E. u E. v ( t = <. <. w , u >. , v >. /\ ch ) }
14 11 12 13 3eqtr4i 
 |-  { <. <. x , y >. , z >. | ps } = { <. <. w , u >. , v >. | ch }