Theorem iotabidv 5211

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

Step	Hyp	Ref	Expression
1		iotabidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	alrimiv 1884	. 2 |- ( ph -> A. x ( ps <-> ch ) )
3		iotabi 5199	. 2 |- ( A. x ( ps <-> ch ) -> ( iota x ps ) = ( iota x ch ) )
4	2, 3	syl 14	1 |- ( ph -> ( iota x ps ) = ( iota x ch ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1361   = wceq 1363   iotacio 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-uni 3822  df-iota 5190

This theorem is referenced by:  csbiotag  5221  dffv3g  5523  fveq1  5526  fveq2  5527  fvres  5551  csbfv12g  5564  fvco2  5598  riotaeqdv  5845  riotabidv  5846  riotabidva  5860  ovtposg  6274  shftval  10848  sumeq1  11377  sumeq2  11381  zsumdc  11406  isumclim3  11445  isumshft  11512  prodeq1f  11574  prodeq2w  11578  prodeq2  11579  zproddc  11601  pcval  12310  grpidvalg  12811  grpidpropdg  12812  dfur2g  13214  oppr0g  13329  oppr1g  13330