Theorem opabbidv 4149

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 1574	. 2 |- F/ x ph
2		nfv 1574	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 4148	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   {copab 4143

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-opab 4145

This theorem is referenced by:  opabbii  4150  csbopabg  4161  xpeq1  4732  xpeq2  4733  opabbi2dv  4870  csbcnvg  4905  resopab2  5051  mptcnv  5130  cores  5231  xpcom  5274  dffn5im  5678  f1oiso2  5950  f1ocnvd  6206  ofreq  6220  f1od2  6379  shftfvalg  11324  shftfval  11327  2shfti  11337  prdsex  13297  prdsval  13301  releqgg  13752  eqgex  13753  eqgfval  13754  reldvdsrsrg  14050  dvdsrvald  14051  dvdsrpropdg  14105  aprval  14240  aprap  14244  lmfval  14860  lgsquadlem3  15752  wksfval  16028