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Theorem riotabidv 5610
Description: Formula-building deduction 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 biidd 170 . . . 4  |-  ( ph  ->  ( x  e.  A  <->  x  e.  A ) )
2 riotabidv.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2anbi12d 457 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
43iotabidv 5001 . 2  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  A  /\  ch ) ) )
5 df-riota 5608 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
6 df-riota 5608 . 2  |-  ( iota_ x  e.  A  ch )  =  ( iota x
( x  e.  A  /\  ch ) )
74, 5, 63eqtr4g 2145 1  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  A  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   iotacio 4978   iota_crio 5607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-uni 3654  df-iota 4980  df-riota 5608
This theorem is referenced by:  riotaeqbidv  5611  csbriotag  5620  infvalti  6717  caucvgsrlemfv  7336  axcaucvglemval  7432  axcaucvglemcau  7433  subval  7674  divvalap  8141  divfnzn  9106  flval  9679  cjval  10279  sqrtrval  10433  qnumval  11441  qdenval  11442
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