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Metamath Proof Explorer

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 } )