Theorem iotabidv 5199

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 1874	. 2 |- ( ph -> A. x ( ps <-> ch ) )
3		iotabi 5187	. 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 1351   = wceq 1353   iotacio 5176

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-uni 3810  df-iota 5178

This theorem is referenced by:  csbiotag  5209  dffv3g  5511  fveq1  5514  fveq2  5515  fvres  5539  csbfv12g  5551  fvco2  5585  riotaeqdv  5831  riotabidv  5832  riotabidva  5846  ovtposg  6259  shftval  10833  sumeq1  11362  sumeq2  11366  zsumdc  11391  isumclim3  11430  isumshft  11497  prodeq1f  11559  prodeq2w  11563  prodeq2  11564  zproddc  11586  pcval  12295  grpidvalg  12791  grpidpropdg  12792  dfur2g  13143  oppr0g  13249  oppr1g  13250