Theorem opabbidv 4087

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 1539	. 2 |- F/ x ph
2		nfv 1539	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 4086	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   {copab 4081

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-opab 4083

This theorem is referenced by:  opabbii  4088  csbopabg  4099  xpeq1  4661  xpeq2  4662  opabbi2dv  4797  csbcnvg  4832  resopab2  4975  mptcnv  5052  cores  5153  xpcom  5196  dffn5im  5585  f1oiso2  5852  f1ocnvd  6100  ofreq  6114  f1od2  6264  shftfvalg  10868  shftfval  10871  2shfti  10881  prdsex  12785  releqgg  13184  eqgex  13185  eqgfval  13186  reldvdsrsrg  13467  dvdsrvald  13468  dvdsrpropdg  13522  aprval  13623  aprap  13627  lmfval  14177