Theorem opabbid 4001

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.)

Hypotheses

Ref	Expression
opabbid.1	|- F/ x ph
opabbid.2	|- F/ y ph
opabbid.3	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
opabbid	|- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Proof of Theorem opabbid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabbid.1	. . . 4 |- F/ x ph
2		opabbid.2	. . . . 5 |- F/ y ph
3		opabbid.3	. . . . . 6 |- ( ph -> ( ps <-> ch ) )
4	3	anbi2d 460	. . . . 5 |- ( ph -> ( ( z = <. x , y >. /\ ps ) <-> ( z = <. x , y >. /\ ch ) ) )
5	2, 4	exbid 1596	. . . 4 |- ( ph -> ( E. y ( z = <. x , y >. /\ ps ) <-> E. y ( z = <. x , y >. /\ ch ) ) )
6	1, 5	exbid 1596	. . 3 |- ( ph -> ( E. x E. y ( z = <. x , y >. /\ ps ) <-> E. x E. y ( z = <. x , y >. /\ ch ) ) )
7	6	abbidv 2258	. 2 |- ( ph -> { z | E. x E. y ( z = <. x , y >. /\ ps ) } = { z | E. x E. y ( z = <. x , y >. /\ ch ) } )
8		df-opab 3998	. 2 |- { <. x , y >. | ps } = { z | E. x E. y ( z = <. x , y >. /\ ps ) }
9		df-opab 3998	. 2 |- { <. x , y >. | ch } = { z | E. x E. y ( z = <. x , y >. /\ ch ) }
10	7, 8, 9	3eqtr4g 2198	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   F/wnf 1437   E.wex 1469   {cab 2126   <.cop 3535   {copab 3996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-opab 3998

This theorem is referenced by:  opabbidv  4002  mpteq12f  4016  fnoprabg  5880