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 }