Theorem opabbidv 4160

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 4159	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   {copab 4154

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-opab 4156

This theorem is referenced by:  opabbii  4161  csbopabg  4172  xpeq1  4745  xpeq2  4746  opabbi2dv  4885  csbcnvg  4920  resopab2  5066  mptcnv  5146  cores  5247  xpcom  5290  dffn5im  5700  f1oiso2  5978  f1ocnvd  6235  ofreq  6248  f1od2  6409  shftfvalg  11441  shftfval  11444  2shfti  11454  prdsex  13415  prdsval  13419  releqgg  13870  eqgex  13871  eqgfval  13872  dvdsrvald  14171  dvdsrpropdg  14225  aprval  14361  aprap  14365  lmfval  14987  lgsquadlem3  15881  wksfval  16246  trlsfvalg  16307  eupthsg  16369