Theorem riotabidv 5900

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 5253	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. A /\ ch ) ) )
5		df-riota 5898	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 5898	. 2 |- ( iota_ x e. A ch ) = ( iota x ( x e. A /\ ch ) )
7	4, 5, 6	3eqtr4g 2262	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   iotacio 5229   iota_crio 5897

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-uni 3850  df-iota 5231  df-riota 5898

This theorem is referenced by:  riotaeqbidv  5901  csbriotag  5911  infvalti  7123  caucvgsrlemfv  7903  axcaucvglemval  8009  axcaucvglemcau  8010  subval  8263  divvalap  8746  divfnzn  9741  flval  10413  cjval  11127  sqrtrval  11282  qnumval  12478  qdenval  12479  grpinvval  13346