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  2955  difjust  3131  unjust  3133  injust  3135  uniiunlem  3245  dfif3  3548  pwjust  3577  snjust  3598  intab  3874  iotajust  5178  tfrlemi1  6333  tfr1onlemaccex  6349  tfrcllemaccex  6362  frecsuc  6408  isbth  6966  nqprlu  7546  recexpr  7637  caucvgprprlemval  7687  caucvgprprlemnbj  7692  caucvgprprlemaddq  7707  caucvgprprlem1  7708  caucvgprprlem2  7709  axcaucvg  7899  mertensabs  11545  bds  14606