Theorem cbvabv 2206

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 1462	. 2 |- F/ y ph
2		nfv 1462	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2205	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   {cab 2069

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076

This theorem is referenced by:  cdeqab1  2818  difjust  2985  unjust  2987  injust  2989  uniiunlem  3093  dfif3  3386  pwjust  3407  snjust  3427  intab  3691  iotajust  4933  tfrlemi1  6029  tfr1onlemaccex  6045  tfrcllemaccex  6058  frecsuc  6104  nqprlu  7009  recexpr  7100  caucvgprprlemval  7150  caucvgprprlemnbj  7155  caucvgprprlemaddq  7170  caucvgprprlem1  7171  caucvgprprlem2  7172  axcaucvg  7338  bds  11085