Theorem opabbidv 4192

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 1581	. 2 |- F/ x ph
2		nfv 1581	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 4191	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   {copab 4186

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-opab 4188

This theorem is referenced by:  opabbii  4193  csbopabg  4204  xpeq1  4783  xpeq2  4784  opabbi2dv  4924  csbcnvg  4959  resopab2  5105  mptcnv  5185  cores  5286  xpcom  5329  dffn5im  5742  f1oiso2  6023  f1ocnvd  6282  f1o3d  6288  ofreq  6296  f1od2  6461  shftfvalg  11561  shftfval  11564  2shfti  11574  releqgg  14000  eqgex  14001  eqgfval  14002  prdsex  14149  prdsval  14150  dvdsrvald  14373  dvdsrpropdg  14427  aprval  14564  aprap  14571  aprprop  14574  lmfval  15217  lgsquadlem3  16112  wksfval  16477  trlsfvalg  16538  eupthsg  16600