Theorem riotabidv 5955

Description: Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)

Hypothesis

Ref	Expression
riotabidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
riotabidv	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem riotabidv

Step	Hyp	Ref	Expression
1		biidd 172	. . . 4 |- ( ph -> ( x e. A <-> x e. A ) )
2		riotabidv.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
3	1, 2	anbi12d 473	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	3	iotabidv 5300	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. A /\ ch ) ) )
5		df-riota 5953	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 5953	. 2 |- ( iota_ x e. A ch ) = ( iota x ( x e. A /\ ch ) )
7	4, 5, 6	3eqtr4g 2287	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   iotacio 5275   iota_crio 5952

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  df-riota 5953

This theorem is referenced by:  riotaeqbidv  5956  csbriotag  5967  infvalti  7185  caucvgsrlemfv  7974  axcaucvglemval  8080  axcaucvglemcau  8081  subval  8334  divvalap  8817  divfnzn  9812  flval  10487  cjval  11351  sqrtrval  11506  qnumval  12702  qdenval  12703  grpinvval  13571  uspgredg2v  16013  usgredg2v  16016