Theorem opabbidv 4071

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 1528	. 2 |- F/ x ph
2		nfv 1528	. 2 |- F/ y ph
3		opabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 2, 3	opabbid 4070	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   {copab 4065

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-opab 4067

This theorem is referenced by:  opabbii  4072  csbopabg  4083  xpeq1  4642  xpeq2  4643  opabbi2dv  4778  csbcnvg  4813  resopab2  4956  mptcnv  5033  cores  5134  xpcom  5177  dffn5im  5563  f1oiso2  5830  f1ocnvd  6075  ofreq  6088  f1od2  6238  shftfvalg  10829  shftfval  10832  2shfti  10842  prdsex  12723  releqgg  13085  eqgfval  13086  reldvdsrsrg  13266  dvdsrvald  13267  dvdsrpropdg  13321  aprval  13377  aprap  13381  lmfval  13731