Theorem iotabidv 5300

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 1920	. 2 |- ( ph -> A. x ( ps <-> ch ) )
3		iotabi 5287	. 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 1393   = wceq 1395   iotacio 5275

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3888  df-iota 5277

This theorem is referenced by:  csbiotag  5310  dffv3g  5622  fveq1  5625  fveq2  5626  fvres  5650  csbfv12g  5666  fvco2  5702  riotaeqdv  5954  riotabidv  5955  riotabidva  5971  ovtposg  6403  shftval  11331  sumeq1  11861  sumeq2  11865  zsumdc  11890  isumclim3  11929  isumshft  11996  prodeq1f  12058  prodeq2w  12062  prodeq2  12063  zproddc  12085  pcval  12814  grpidvalg  13401  grpidpropdg  13402  igsumvalx  13417  gsumpropd  13420  gsumpropd2  13421  gsumress  13423  gsumval2  13425  dfur2g  13920  oppr0g  14039  oppr1g  14040