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  2954  difjust  3130  unjust  3132  injust  3134  uniiunlem  3244  dfif3  3547  pwjust  3576  snjust  3597  intab  3873  iotajust  5177  tfrlemi1  6332  tfr1onlemaccex  6348  tfrcllemaccex  6361  frecsuc  6407  isbth  6965  nqprlu  7545  recexpr  7636  caucvgprprlemval  7686  caucvgprprlemnbj  7691  caucvgprprlemaddq  7706  caucvgprprlem1  7707  caucvgprprlem2  7708  axcaucvg  7898  mertensabs  11544  bds  14573