Theorem cbvabv 2262

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 1508	. 2 |- F/ y ph
2		nfv 1508	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2261	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   {cab 2123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130

This theorem is referenced by:  cdeqab1  2896  difjust  3067  unjust  3069  injust  3071  uniiunlem  3180  dfif3  3482  pwjust  3506  snjust  3527  intab  3795  iotajust  5082  tfrlemi1  6222  tfr1onlemaccex  6238  tfrcllemaccex  6251  frecsuc  6297  isbth  6848  nqprlu  7348  recexpr  7439  caucvgprprlemval  7489  caucvgprprlemnbj  7494  caucvgprprlemaddq  7509  caucvgprprlem1  7510  caucvgprprlem2  7511  axcaucvg  7701  mertensabs  11299  bds  13038