Theorem iotabidv 5301

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 5288	. 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 5276

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 3889  df-iota 5278

This theorem is referenced by:  csbiotag  5311  dffv3g  5625  fveq1  5628  fveq2  5629  fvres  5653  csbfv12g  5669  fvco2  5705  riotaeqdv  5961  riotabidv  5962  riotabidva  5978  ovtposg  6411  shftval  11351  sumeq1  11881  sumeq2  11885  zsumdc  11910  isumclim3  11949  isumshft  12016  prodeq1f  12078  prodeq2w  12082  prodeq2  12083  zproddc  12105  pcval  12834  grpidvalg  13421  grpidpropdg  13422  igsumvalx  13437  gsumpropd  13440  gsumpropd2  13441  gsumress  13443  gsumval2  13445  dfur2g  13940  oppr0g  14059  oppr1g  14060