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Theorem iotabidv 5169
Description: Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
Hypothesis
Ref Expression
iotabidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
iotabidv  |-  ( ph  ->  ( iota x ps )  =  ( iota
x ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem iotabidv
StepHypRef Expression
1 iotabidv.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21alrimiv 1861 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
3 iotabi 5157 . 2  |-  ( A. x ( ps  <->  ch )  ->  ( iota x ps )  =  ( iota
x ch ) )
42, 3syl 14 1  |-  ( ph  ->  ( iota x ps )  =  ( iota
x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1340    = wceq 1342   iotacio 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-uni 3785  df-iota 5148
This theorem is referenced by:  csbiotag  5176  dffv3g  5477  fveq1  5480  fveq2  5481  fvres  5505  csbfv12g  5517  fvco2  5550  riotaeqdv  5794  riotabidv  5795  riotabidva  5809  ovtposg  6219  shftval  10757  sumeq1  11286  sumeq2  11290  zsumdc  11315  isumclim3  11354  isumshft  11421  prodeq1f  11483  prodeq2w  11487  prodeq2  11488  zproddc  11510  pcval  12217
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