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Theorem iotabidv 5256
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 1621 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
3 iotabi 5244 . 2  |-  ( A. x ( ps  <->  ch )  ->  ( iota x ps )  =  ( iota
x ch ) )
42, 3syl 15 1  |-  ( ph  ->  ( iota x ps )  =  ( iota
x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    = wceq 1632   iotacio 5233
This theorem is referenced by:  csbiotag  5264  dffv3  5537  fveq1  5540  fveq2  5541  csbfv12g  5551  fvres  5558  fvco2  5610  fvopab5  6305  opabiota  6309  riotaeqdv  6321  riotabidv  6322  riotabidva  6337  erov  6771  iunfictbso  7757  isf32lem9  8003  shftval  11585  sumeq1f  12177  sumeq2w  12181  sumeq2ii  12182  cbvsum  12184  zsum  12207  isumclim3  12238  isumshft  12314  pcval  12913  grpidval  14400  grpidpropd  14415  gsumvalx  14467  gsumpropd  14469  gsumress  14470  dchrptlem1  20519  lgsdchrval  20602  ajval  21456  adjval  22486  gsumpropd2lem  23394  cprodeq1f  24130  cprodeq2w  24134  cprodeq2ii  24135  zprod  24160  psgnfval  27526  psgnval  27533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-uni 3844  df-iota 5235
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