Theorem oprabbidv

Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004)

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

Proof

Step	Hyp	Ref	Expression
1		oprabbidv.1	|- ( ph -> ( ps <-> ch ) )
2	1	anbi2d	|- ( ph -> ( ( w = <. <. x , y >. , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ch ) ) )
3	2	exbidv	|- ( ph -> ( E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
4	3	exbidv	|- ( ph -> ( E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
5	4	exbidv	|- ( ph -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
6	5	abbidv	|- ( ph -> { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) } )
7		df-oprab	|- { <. <. x , y >. , z >. | ps } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) }
8		df-oprab	|- { <. <. x , y >. , z >. | ch } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) }
9	6 7 8	3eqtr4g	|- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )