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Theorem iotabidv 5398
Description: Formula-building deduction rule 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 1638 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
3 iotabi 5386 . 2  |-  ( A. x ( ps  <->  ch )  ->  ( iota x ps )  =  ( iota
x ch ) )
42, 3syl 16 1  |-  ( ph  ->  ( iota x ps )  =  ( iota
x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649   iotacio 5375
This theorem is referenced by:  csbiotag  5406  dffv3  5683  fveq1  5686  fveq2  5687  csbfv12g  5697  fvres  5704  fvco2  5757  fvopab5  6493  opabiota  6497  riotaeqdv  6509  riotabidv  6510  riotabidva  6525  erov  6960  iunfictbso  7951  isf32lem9  8197  shftval  11844  sumeq1f  12437  sumeq2w  12441  sumeq2ii  12442  cbvsum  12444  zsum  12467  isumclim3  12498  isumshft  12574  pcval  13173  grpidval  14662  grpidpropd  14677  gsumvalx  14729  gsumpropd  14731  gsumress  14732  dchrptlem1  21001  lgsdchrval  21084  ajval  22316  adjval  23346  gsumpropd2lem  24173  prodeq1f  25187  prodeq2w  25191  prodeq2ii  25192  zprod  25216  iprodclim3  25266  psgnfval  27291  psgnval  27298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-uni 3976  df-iota 5377
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