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Theorem iotabidv 5570
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 1695 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
3 iotabi 5558 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379   iotacio 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-uni 4246  df-iota 5549
This theorem is referenced by:  csbiota  5578  csbiotagOLD  5579  dffv3  5860  fveq1  5863  fveq2  5864  fvres  5878  csbfv12  5899  csbfv12gOLD  5900  opabiota  5928  fvco2  5940  fvopab5  5971  riotaeqdv  6244  riotabidv  6245  riotabidva  6260  erov  7405  iunfictbso  8491  isf32lem9  8737  shftval  12866  sumeq1  13470  sumeq2w  13473  sumeq2ii  13474  zsum  13499  isumclim3  13533  isumshft  13610  pcval  14223  grpidval  15745  grpidpropd  15760  gsumvalx  15815  gsumpropd  15817  gsumpropd2lem  15818  gsumress  15820  psgnfval  16321  psgnval  16328  psgndif  18405  dchrptlem1  23267  lgsdchrval  23350  ajval  25453  adjval  26485  resv1r  27490  prodeq1f  28617  prodeq2w  28621  prodeq2ii  28622  zprod  28646  iprodclim3  28696  bj-finsumval0  33735
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