Theorem opabbidv 4081

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 1538	. 2 |- F/ x ph
2		nfv 1538	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 4080	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   {copab 4075

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-opab 4077

This theorem is referenced by:  opabbii  4082  csbopabg  4093  xpeq1  4652  xpeq2  4653  opabbi2dv  4788  csbcnvg  4823  resopab2  4966  mptcnv  5043  cores  5144  xpcom  5187  dffn5im  5574  f1oiso2  5841  f1ocnvd  6086  ofreq  6099  f1od2  6249  shftfvalg  10840  shftfval  10843  2shfti  10853  prdsex  12735  releqgg  13109  eqgex  13110  eqgfval  13111  reldvdsrsrg  13324  dvdsrvald  13325  dvdsrpropdg  13379  aprval  13435  aprap  13439  lmfval  13932