Theorem iotabidv 5355

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 1927	. 2 |- ( ph -> A. x ( ps <-> ch ) )
3		iotabi 5342	. 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 1400   = wceq 1402   iotacio 5330

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3931  df-iota 5332

This theorem is referenced by:  csbiotag  5365  dffv3g  5686  fveq1  5689  fveq2  5690  fvres  5714  csbfv12g  5730  fvco2  5768  riotaeqdv  6029  riotabidv  6030  riotabidva  6046  ovtposg  6520  shftval  11568  sumeq1  12099  sumeq2  12103  zsumdc  12129  isumclim3  12168  isumshft  12235  prodeq1f  12297  prodeq2w  12301  prodeq2  12302  zproddc  12324  pcval  13053  grpidvalg  13670  grpidpropdg  13671  gzsumvalx  13686  gzsumress  13689  gzsumval2  13691  gsumvalfi  14129  dfur2g  14240  oppr0g  14360  oppr1g  14361