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Theorem iotabidv 5079
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 1830 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
3 iotabi 5067 . 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 1314    = wceq 1316   iotacio 5056
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-uni 3707  df-iota 5058
This theorem is referenced by:  csbiotag  5086  dffv3g  5385  fveq1  5388  fveq2  5389  fvres  5413  csbfv12g  5425  fvco2  5458  riotaeqdv  5699  riotabidv  5700  riotabidva  5714  ovtposg  6124  shftval  10565  sumeq1  11092  sumeq2  11096  zsumdc  11121  isumclim3  11160  isumshft  11227
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