MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvrabv Unicode version

Theorem cbvrabv 2915
Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
Hypothesis
Ref Expression
cbvrabv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrabv  |-  { x  e.  A  |  ph }  =  { y  e.  A  |  ps }
Distinct variable groups:    x, y, A    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem cbvrabv
StepHypRef Expression
1 nfcv 2540 . 2  |-  F/_ x A
2 nfcv 2540 . 2  |-  F/_ y A
3 nfv 1626 . 2  |-  F/ y
ph
4 nfv 1626 . 2  |-  F/ x ps
5 cbvrabv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
61, 2, 3, 4, 5cbvrab 2914 1  |-  { x  e.  A  |  ph }  =  { y  e.  A  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   {crab 2670
This theorem is referenced by:  pwnss  4325  knatar  6039  oeeulem  6803  ordtypecbv  7442  ordtypelem9  7451  inf3lema  7535  oemapso  7594  oemapvali  7596  tz9.12lem3  7671  cofsmo  8105  enfin2i  8157  fin23lem33  8181  isf32lem11  8199  zorn2g  8339  pwfseqlem1  8489  pwfseqlem3  8491  zsupss  10521  zmin  10526  reexALT  10562  hashbc  11657  sqrlem7  12009  rpnnen  12781  divalglem5  12872  bitsfzolem  12901  smupp1  12947  gcdcllem3  12968  bezout  12997  eulerth  13127  odzval  13132  pcprecl  13168  pcprendvds  13169  pcpremul  13172  pceulem  13174  prmreclem1  13239  prmreclem5  13243  prmreclem6  13244  4sqlem19  13286  vdwnn  13321  hashbcval  13325  gsumvalx  14729  efgsdm  15317  efgsfo  15326  ablfaclem3  15600  ltbwe  16488  coe1mul2lem2  16616  pptbas  17027  concompss  17449  ptcmplem5  18040  ustuqtop  18229  utopsnneip  18231  icccmplem2  18807  minveclem5  19287  ivth  19304  ovolicc2lem5  19370  ovolicc  19372  opnmbllem  19446  vitali  19458  itg2monolem3  19597  elqaa  20192  radcnvle  20289  pserdvlem2  20297  wilth  20807  ftalem6  20813  nbgraf1olem1  21404  nbgraf1o  21410  cusgraexg  21431  cusgrafi  21444  ubthlem3  22327  htth  22374  ballotlemelo  24698  ballotleme  24707  ballotlem7  24746  lgamgulmlem5  24770  lgamcvglem  24777  subfacp1lem6  24824  erdsze  24841  cvmscbv  24898  cvmsiota  24917  cvmlift2lem13  24955  axcontlem11  25817  mblfinlem  26143  neibastop2  26280  eldioph4i  26763  stoweidlem49  27665  frgraregorufr0  28155  usgreghash2spot  28172  lclkrs2  32023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675
  Copyright terms: Public domain W3C validator