Theorem opabbid

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004) (Proof shortened by Andrew Salmon, 9-Jul-2011)

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

Proof

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