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Theorem riotabidv 6510
Description: Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotabidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
riotabidv  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem riotabidv
StepHypRef Expression
1 riotabidv.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
21reubidv 2852 . . 3  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)
31anbi2d 685 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
43iotabidv 5398 . . 3  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  A  /\  ch ) ) )
5 eqidd 2405 . . 3  |-  ( ph  ->  ( Undef `  { x  |  x  e.  A } )  =  (
Undef `  { x  |  x  e.  A }
) )
62, 4, 5ifbieq12d 3721 . 2  |-  ( ph  ->  if ( E! x  e.  A  ps ,  ( iota x ( x  e.  A  /\  ps ) ) ,  (
Undef `  { x  |  x  e.  A }
) )  =  if ( E! x  e.  A  ch ,  ( iota x ( x  e.  A  /\  ch ) ) ,  (
Undef `  { x  |  x  e.  A }
) ) )
7 df-riota 6508 . 2  |-  ( iota_ x  e.  A ps )  =  if ( E! x  e.  A  ps ,  ( iota x ( x  e.  A  /\  ps ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
8 df-riota 6508 . 2  |-  ( iota_ x  e.  A ch )  =  if ( E! x  e.  A  ch ,  ( iota x ( x  e.  A  /\  ch ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
96, 7, 83eqtr4g 2461 1  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   E!wreu 2668   ifcif 3699   iotacio 5375   ` cfv 5413   Undefcund 6500   iota_crio 6501
This theorem is referenced by:  riotaeqbidv  6511  csbriotag  6521  fin23lem27  8164  subval  9253  divval  9636  flval  11158  cjval  11862  sqrval  11997  qnumval  13084  qdenval  13085  lubval  14391  glbval  14396  joinval2  14401  meetval2  14408  spwval2  14611  spwpr4  14618  grpinvval  14799  pj1fval  15281  pj1val  15282  q1pval  20029  coeval  20095  quotval  20162  usgraidx2v  21365  nbgraf1olem4  21407  grpoinvval  21766  pjhval  22852  nmopadjlei  23544  cdj3lem2  23891  cvmliftlem15  24938  cvmlift2lem4  24946  cvmlift2  24956  cvmlift3lem2  24960  cvmlift3lem4  24962  cvmlift3lem6  24964  cvmlift3lem7  24965  cvmlift3lem9  24967  cvmlift3  24968  fvtransport  25870  unxpwdom3  27124  mpaaval  27224  frgra2v  28103  frgrancvvdeqlemB  28141  frgrancvvdeqlemC  28142  lshpkrlem1  29593  lshpkrlem2  29594  lshpkrlem3  29595  lshpkrcl  29599  trlset  30643  trlval  30644  cdleme27b  30850  cdleme29b  30857  cdleme31so  30861  cdleme31sn1  30863  cdleme31sn1c  30870  cdleme31fv  30872  cdlemefrs29clN  30881  cdleme40v  30951  cdlemg1cN  31069  cdlemg1cex  31070  cdlemksv  31326  cdlemkuu  31377  cdlemkid3N  31415  cdlemkid4  31416  cdlemm10N  31601  dicval  31659  dihval  31715  dochfl1  31959  lcfl7N  31984  lcfrlem8  32032  lcfrlem9  32033  lcf1o  32034  mapdhval  32207  hvmapval  32243  hvmapvalvalN  32244  hdmap1fval  32280  hdmap1vallem  32281  hdmap1val  32282  hdmap1cbv  32286  hdmapfval  32313  hdmapval  32314  hgmapffval  32371  hgmapfval  32372  hgmapval  32373
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-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-un 3285  df-if 3700  df-uni 3976  df-iota 5377  df-riota 6508
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