Theorem opabbidv 4084

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 4083	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   {copab 4078

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 4080

This theorem is referenced by:  opabbii  4085  csbopabg  4096  xpeq1  4655  xpeq2  4656  opabbi2dv  4791  csbcnvg  4826  resopab2  4969  mptcnv  5046  cores  5147  xpcom  5190  dffn5im  5577  f1oiso2  5844  f1ocnvd  6091  ofreq  6104  f1od2  6254  shftfvalg  10845  shftfval  10848  2shfti  10858  prdsex  12740  releqgg  13125  eqgex  13126  eqgfval  13127  reldvdsrsrg  13403  dvdsrvald  13404  dvdsrpropdg  13458  aprval  13559  aprap  13563  lmfval  14089