Theorem riotabidv 5972

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 5309	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. A /\ ch ) ) )
5		df-riota 5970	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 5970	. 2 |- ( iota_ x e. A ch ) = ( iota x ( x e. A /\ ch ) )
7	4, 5, 6	3eqtr4g 2289	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   iotacio 5284   iota_crio 5969

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-uni 3894  df-iota 5286  df-riota 5970

This theorem is referenced by:  riotaeqbidv  5973  csbriotag  5984  infvalti  7220  caucvgsrlemfv  8010  axcaucvglemval  8116  axcaucvglemcau  8117  subval  8370  divvalap  8853  divfnzn  9854  flval  10531  cjval  11405  sqrtrval  11560  qnumval  12756  qdenval  12757  grpinvval  13625  uspgredg2v  16071  usgredg2v  16074