Theorem cbvabv 2265

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)

Hypothesis

Ref	Expression
cbvabv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvabv	|- { x | ph } = { y | ps }

Distinct variable groups:   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvabv

Step	Hyp	Ref	Expression
1		nfv 1509	. 2 |- F/ y ph
2		nfv 1509	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2264	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   {cab 2126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133

This theorem is referenced by:  cdeqab1  2905  difjust  3077  unjust  3079  injust  3081  uniiunlem  3190  dfif3  3492  pwjust  3516  snjust  3537  intab  3808  iotajust  5095  tfrlemi1  6237  tfr1onlemaccex  6253  tfrcllemaccex  6266  frecsuc  6312  isbth  6863  nqprlu  7379  recexpr  7470  caucvgprprlemval  7520  caucvgprprlemnbj  7525  caucvgprprlemaddq  7540  caucvgprprlem1  7541  caucvgprprlem2  7542  axcaucvg  7732  mertensabs  11338  bds  13220