Theorem oprabbid

Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004) (Revised by Mario Carneiro, 24-Jun-2014)

	Ref	Expression
Hypotheses	oprabbid.1	|- F/ x ph
	oprabbid.2	|- F/ y ph
	oprabbid.3	|- F/ z ph
	oprabbid.4	|- ( ph -> ( ps <-> ch ) )
Assertion	oprabbid	|- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Proof

Step	Hyp	Ref	Expression
1		oprabbid.1	|- F/ x ph
2		oprabbid.2	|- F/ y ph
3		oprabbid.3	|- F/ z ph
4		oprabbid.4	|- ( ph -> ( ps <-> ch ) )
5	4	anbi2d	|- ( ph -> ( ( w = <. <. x , y >. , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ch ) ) )
6	3 5	exbid	|- ( ph -> ( E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
7	2 6	exbid	|- ( ph -> ( E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
8	1 7	exbid	|- ( ph -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
9	8	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 ) } )
10		df-oprab	|- { <. <. x , y >. , z >. | ps } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) }
11		df-oprab	|- { <. <. x , y >. , z >. | ch } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) }
12	9 10 11	3eqtr4g	|- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )