Theorem opabbidv 4176

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

Hypothesis

Ref	Expression
opabbidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   ps( x, y)   ch( x, y)

Proof of Theorem opabbidv

Step	Hyp	Ref	Expression
1		nfv 1577	. 2 |- F/ x ph
2		nfv 1577	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 4175	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   {copab 4170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-opab 4172

This theorem is referenced by:  opabbii  4177  csbopabg  4188  xpeq1  4763  xpeq2  4764  opabbi2dv  4904  csbcnvg  4939  resopab2  5085  mptcnv  5165  cores  5266  xpcom  5309  dffn5im  5722  f1oiso2  6000  f1ocnvd  6257  ofreq  6270  f1od2  6431  shftfvalg  11503  shftfval  11506  2shfti  11516  prdsex  13482  prdsval  13486  releqgg  13937  eqgex  13938  eqgfval  13939  dvdsrvald  14238  dvdsrpropdg  14292  aprval  14428  aprap  14432  lmfval  15058  lgsquadlem3  15952  wksfval  16317  trlsfvalg  16378  eupthsg  16440