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Theorem riotabidv 5832
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 172 . . . 4  |-  ( ph  ->  ( x  e.  A  <->  x  e.  A ) )
2 riotabidv.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2anbi12d 473 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
43iotabidv 5199 . 2  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  A  /\  ch ) ) )
5 df-riota 5830 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
6 df-riota 5830 . 2  |-  ( iota_ x  e.  A  ch )  =  ( iota x
( x  e.  A  /\  ch ) )
74, 5, 63eqtr4g 2235 1  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  A  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   iotacio 5176   iota_crio 5829
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-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-uni 3810  df-iota 5178  df-riota 5830
This theorem is referenced by:  riotaeqbidv  5833  csbriotag  5842  infvalti  7020  caucvgsrlemfv  7789  axcaucvglemval  7895  axcaucvglemcau  7896  subval  8148  divvalap  8630  divfnzn  9620  flval  10271  cjval  10853  sqrtrval  11008  qnumval  12184  qdenval  12185  grpinvval  12915
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