Theorem opabbidv 3852

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (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 1462	. 2 |- F/ x ph
2		nfv 1462	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 3851	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   {copab 3846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-opab 3848

This theorem is referenced by:  opabbii  3853  csbopabg  3864  xpeq1  4385  xpeq2  4386  opabbi2dv  4513  csbcnvg  4547  resopab2  4685  cores  4854  xpcom  4894  dffn5im  5251  f1oiso2  5497  f1ocnvd  5733  ofreq  5746  f1od2  5887  sprmpt2  5891  shftfvalg  9844  shftfval  9847  2shfti  9857