Theorem cbvabv 2302

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 1528	. 2 |- F/ y ph
2		nfv 1528	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2301	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   {cab 2163

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170

This theorem is referenced by:  cdeqab1  2954  difjust  3130  unjust  3132  injust  3134  uniiunlem  3244  dfif3  3547  pwjust  3576  snjust  3597  intab  3872  iotajust  5174  tfrlemi1  6328  tfr1onlemaccex  6344  tfrcllemaccex  6357  frecsuc  6403  isbth  6961  nqprlu  7541  recexpr  7632  caucvgprprlemval  7682  caucvgprprlemnbj  7687  caucvgprprlemaddq  7702  caucvgprprlem1  7703  caucvgprprlem2  7704  axcaucvg  7894  mertensabs  11536  bds  14374