Theorem opabbidv 4150

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 4149	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   {copab 4144

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 4146

This theorem is referenced by:  opabbii  4151  csbopabg  4162  xpeq1  4733  xpeq2  4734  opabbi2dv  4871  csbcnvg  4906  resopab2  5052  mptcnv  5131  cores  5232  xpcom  5275  dffn5im  5681  f1oiso2  5957  f1ocnvd  6214  ofreq  6228  f1od2  6387  shftfvalg  11344  shftfval  11347  2shfti  11357  prdsex  13317  prdsval  13321  releqgg  13772  eqgex  13773  eqgfval  13774  dvdsrvald  14072  dvdsrpropdg  14126  aprval  14261  aprap  14265  lmfval  14882  lgsquadlem3  15773  wksfval  16063  trlsfvalg  16122