Theorem opabbidv 4109

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 1550	. 2 |- F/ x ph
2		nfv 1550	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 4108	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   {copab 4103

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-opab 4105

This theorem is referenced by:  opabbii  4110  csbopabg  4121  xpeq1  4688  xpeq2  4689  opabbi2dv  4826  csbcnvg  4861  resopab2  5005  mptcnv  5084  cores  5185  xpcom  5228  dffn5im  5623  f1oiso2  5895  f1ocnvd  6147  ofreq  6161  f1od2  6320  shftfvalg  11071  shftfval  11074  2shfti  11084  prdsex  13043  prdsval  13047  releqgg  13498  eqgex  13499  eqgfval  13500  reldvdsrsrg  13796  dvdsrvald  13797  dvdsrpropdg  13851  aprval  13986  aprap  13990  lmfval  14606  lgsquadlem3  15498