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Theorem iotabidv 5441
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 1642 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
3 iotabi 5429 . 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 178   A.wal 1550    = wceq 1653   iotacio 5418
This theorem is referenced by:  csbiotag  5449  dffv3  5726  fveq1  5729  fveq2  5730  csbfv12g  5740  fvres  5747  fvco2  5800  fvopab5  6536  opabiota  6540  riotaeqdv  6552  riotabidv  6553  riotabidva  6568  erov  7003  iunfictbso  7997  isf32lem9  8243  shftval  11891  sumeq1f  12484  sumeq2w  12488  sumeq2ii  12489  cbvsum  12491  zsum  12514  isumclim3  12545  isumshft  12621  pcval  13220  grpidval  14709  grpidpropd  14724  gsumvalx  14776  gsumpropd  14778  gsumress  14779  dchrptlem1  21050  lgsdchrval  21133  ajval  22365  adjval  23395  gsumpropd2lem  24222  prodeq1f  25236  prodeq2w  25240  prodeq2ii  25241  zprod  25265  iprodclim3  25315  psgnfval  27402  psgnval  27409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-uni 4018  df-iota 5420
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