Theorem cbvabv 2365

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 1581	. 2 |- F/ y ph
2		nfv 1581	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2364	1 |- { x | ph } = { y | ps }

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   {cab 2224

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231

This theorem is used by:  eqabbw  2375  cdeqab1  3043  difjust  3221  unjust  3223  injust  3225  uniiunlem  3338  dfif3  3654  pwjust  3689  snjust  3714  intab  3999  iotajust  5339  cbviotavw  5346  tfrlemi1  6607  tfr1onlemaccex  6623  tfrcllemaccex  6636  frecsuc  6682  isbth  7289  nqprlu  7920  recexpr  8011  caucvgprprlemval  8061  caucvgprprlemnbj  8066  caucvgprprlemaddq  8081  caucvgprprlem1  8082  caucvgprprlem2  8083  axcaucvg  8273  hashf1lem2  11308  mertensabs  12329  4sq  13218  ballotfilemfmpn  13292  isuhgrm  16517  isushgrm  16518  isupgren  16541  isumgren  16551  isuspgren  16603  isusgren  16604  bds  17082