Theorem cbvabv 2361

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 1577	. 2 |- F/ y ph
2		nfv 1577	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2360	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   {cab 2220

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227

This theorem is referenced by:  cdeqab1  3036  difjust  3214  unjust  3216  injust  3218  uniiunlem  3330  dfif3  3638  pwjust  3672  snjust  3696  intab  3980  iotajust  5313  cbviotavw  5320  tfrlemi1  6565  tfr1onlemaccex  6581  tfrcllemaccex  6594  frecsuc  6640  isbth  7239  nqprlu  7864  recexpr  7955  caucvgprprlemval  8005  caucvgprprlemnbj  8010  caucvgprprlemaddq  8025  caucvgprprlem1  8026  caucvgprprlem2  8027  axcaucvg  8217  mertensabs  12227  4sq  13112  ballotfilemfmpn  13155  isuhgrm  16083  isushgrm  16084  isupgren  16107  isumgren  16117  isuspgren  16169  isusgren  16170  bds  16638