Theorem cbvabv 2354

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 1574	. 2 |- F/ y ph
2		nfv 1574	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2353	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   {cab 2215

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222

This theorem is referenced by:  cdeqab1  3021  difjust  3199  unjust  3201  injust  3203  uniiunlem  3314  dfif3  3617  pwjust  3651  snjust  3672  intab  3953  iotajust  5281  cbviotavw  5288  tfrlemi1  6491  tfr1onlemaccex  6507  tfrcllemaccex  6520  frecsuc  6566  isbth  7155  nqprlu  7755  recexpr  7846  caucvgprprlemval  7896  caucvgprprlemnbj  7901  caucvgprprlemaddq  7916  caucvgprprlem1  7917  caucvgprprlem2  7918  axcaucvg  8108  mertensabs  12085  4sq  12970  isuhgrm  15908  isushgrm  15909  isupgren  15932  isumgren  15942  isuspgren  15992  isusgren  15993  bds  16356