Theorem iotabidv 5200

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 5188	. 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 5177

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 3811  df-iota 5179

This theorem is referenced by:  csbiotag  5210  dffv3g  5512  fveq1  5515  fveq2  5516  fvres  5540  csbfv12g  5552  fvco2  5586  riotaeqdv  5832  riotabidv  5833  riotabidva  5847  ovtposg  6260  shftval  10834  sumeq1  11363  sumeq2  11367  zsumdc  11392  isumclim3  11431  isumshft  11498  prodeq1f  11560  prodeq2w  11564  prodeq2  11565  zproddc  11587  pcval  12296  grpidvalg  12792  grpidpropdg  12793  dfur2g  13145  oppr0g  13251  oppr1g  13252