Theorem cbvabv 2208

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 1464	. 2 |- F/ y ph
2		nfv 1464	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2207	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1287   {cab 2071

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078

This theorem is referenced by:  cdeqab1  2821  difjust  2989  unjust  2991  injust  2993  uniiunlem  3098  dfif3  3392  pwjust  3416  snjust  3436  intab  3700  iotajust  4945  tfrlemi1  6051  tfr1onlemaccex  6067  tfrcllemaccex  6080  frecsuc  6126  isbth  6620  nqprlu  7050  recexpr  7141  caucvgprprlemval  7191  caucvgprprlemnbj  7196  caucvgprprlemaddq  7211  caucvgprprlem1  7212  caucvgprprlem2  7213  axcaucvg  7379  bds  11187