Theorem opabbidv

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995)

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

Proof

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