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  6608  tfr1onlemaccex  6624  tfrcllemaccex  6637  frecsuc  6683  isbth  7291  nqprlu  7920  recexpr  8011  caucvgprprlemval  8061  caucvgprprlemnbj  8066  caucvgprprlemaddq  8081  caucvgprprlem1  8082  caucvgprprlem2  8083  axcaucvg  8273  hashf1lem2  11307  mertensabs  12328  4sq  13217  ballotfilemfmpn  13291  isuhgrm  16534  isushgrm  16535  isupgren  16558  isumgren  16568  isuspgren  16620  isusgren  16621  bds  17099