Theorem opabbidv 4066

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 4065	1 |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   {copab 4060

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 4062

This theorem is referenced by:  opabbii  4067  csbopabg  4078  xpeq1  4637  xpeq2  4638  opabbi2dv  4772  csbcnvg  4807  resopab2  4950  mptcnv  5027  cores  5128  xpcom  5171  dffn5im  5557  f1oiso2  5822  f1ocnvd  6067  ofreq  6080  f1od2  6230  shftfvalg  10811  shftfval  10814  2shfti  10824  reldvdsrsrg  13086  dvdsrvald  13087  dvdsrpropdg  13139  lmfval  13359