Theorem opabbidv 4118

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 1552	. 2 |- F/ x ph
2		nfv 1552	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 4117	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   {copab 4112

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-opab 4114

This theorem is referenced by:  opabbii  4119  csbopabg  4130  xpeq1  4697  xpeq2  4698  opabbi2dv  4835  csbcnvg  4870  resopab2  5015  mptcnv  5094  cores  5195  xpcom  5238  dffn5im  5637  f1oiso2  5909  f1ocnvd  6161  ofreq  6175  f1od2  6334  shftfvalg  11204  shftfval  11207  2shfti  11217  prdsex  13176  prdsval  13180  releqgg  13631  eqgex  13632  eqgfval  13633  reldvdsrsrg  13929  dvdsrvald  13930  dvdsrpropdg  13984  aprval  14119  aprap  14123  lmfval  14739  lgsquadlem3  15631