Theorem cbvabv 2161

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 1421	. 2 |- F/ y ph
2		nfv 1421	. 2 |- F/ x ps
3		cbvabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2160	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033

This theorem is referenced by:  cdeqab1  2756  difjust  2919  unjust  2921  injust  2923  uniiunlem  3028  dfif3  3343  pwjust  3360  snjust  3380  intab  3644  iotajust  4866  tfrlemi1  5946  frecsuc  5991  nqprlu  6645  recexpr  6736  caucvgprprlemval  6786  caucvgprprlemnbj  6791  caucvgprprlemaddq  6806  caucvgprprlem1  6807  caucvgprprlem2  6808  axcaucvg  6974  bds  9971