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Metamath Proof Explorer

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 } )