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Theorem riotabidv 5875
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 5237 . 2  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  A  /\  ch ) ) )
5 df-riota 5873 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
6 df-riota 5873 . 2  |-  ( iota_ x  e.  A  ch )  =  ( iota x
( x  e.  A  /\  ch ) )
74, 5, 63eqtr4g 2251 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 1364    e. wcel 2164   iotacio 5213   iota_crio 5872
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-uni 3836  df-iota 5215  df-riota 5873
This theorem is referenced by:  riotaeqbidv  5876  csbriotag  5886  infvalti  7081  caucvgsrlemfv  7851  axcaucvglemval  7957  axcaucvglemcau  7958  subval  8211  divvalap  8693  divfnzn  9686  flval  10341  cjval  10989  sqrtrval  11144  qnumval  12323  qdenval  12324  grpinvval  13115
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