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Theorem riotabidv 5521
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 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 4938 . 2  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  A  /\  ch ) ) )
5 df-riota 5519 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
6 df-riota 5519 . 2  |-  ( iota_ x  e.  A  ch )  =  ( iota x
( x  e.  A  /\  ch ) )
74, 5, 63eqtr4g 2140 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 1285    e. wcel 1434   iotacio 4915   iota_crio 5518
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-uni 3622  df-iota 4917  df-riota 5519
This theorem is referenced by:  riotaeqbidv  5522  csbriotag  5531  infvalti  6529  caucvgsrlemfv  7081  axcaucvglemval  7177  axcaucvglemcau  7178  subval  7419  divvalap  7881  divfnzn  8839  flval  9406  cjval  9933  sqrtrval  10087  qnumval  10770  qdenval  10771
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