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Metamath Proof Explorer

Theorem cbvoprab3v

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004) (Revised by David Abernethy, 19-Jun-2012)

Ref Expression
Hypothesis cbvoprab3v.1 
|- ( z = w -> ( ph <-> ps ) )
Assertion cbvoprab3v 
|- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Proof

Step Hyp Ref Expression
1 cbvoprab3v.1 
 |-  ( z = w -> ( ph <-> ps ) )
2 opeq2 
 |-  ( z = w -> <. <. x , y >. , z >. = <. <. x , y >. , w >. )
3 2 eqeq2d 
 |-  ( z = w -> ( v = <. <. x , y >. , z >. <-> v = <. <. x , y >. , w >. ) )
4 3 1 anbi12d 
 |-  ( z = w -> ( ( v = <. <. x , y >. , z >. /\ ph ) <-> ( v = <. <. x , y >. , w >. /\ ps ) ) )
5 4 cbvexvw 
 |-  ( E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. w ( v = <. <. x , y >. , w >. /\ ps ) )
6 5 2exbii 
 |-  ( E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. w ( v = <. <. x , y >. , w >. /\ ps ) )
7 6 abbii 
 |-  { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) } = { v | E. x E. y E. w ( v = <. <. x , y >. , w >. /\ ps ) }
8 df-oprab 
 |-  { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
9 df-oprab 
 |-  { <. <. x , y >. , w >. | ps } = { v | E. x E. y E. w ( v = <. <. x , y >. , w >. /\ ps ) }
10 7 8 9 3eqtr4i 
 |-  { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }