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  3020  difjust  3198  unjust  3200  injust  3202  uniiunlem  3313  dfif3  3616  pwjust  3650  snjust  3671  intab  3952  iotajust  5277  cbviotavw  5284  tfrlemi1  6478  tfr1onlemaccex  6494  tfrcllemaccex  6507  frecsuc  6553  isbth  7134  nqprlu  7734  recexpr  7825  caucvgprprlemval  7875  caucvgprprlemnbj  7880  caucvgprprlemaddq  7895  caucvgprprlem1  7896  caucvgprprlem2  7897  axcaucvg  8087  mertensabs  12048  4sq  12933  isuhgrm  15871  isushgrm  15872  isupgren  15895  isumgren  15905  isuspgren  15955  isusgren  15956  bds  16214