Theorem riotabidv 6043

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 477	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	3	iotabidv 5363	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. A /\ ch ) ) )
5		df-riota 6041	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 6041	. 2 |- ( iota_ x e. A ch ) = ( iota x ( x e. A /\ ch ) )
7	4, 5, 6	3eqtr4g 2296	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   iotacio 5338   iota_crio 6040

This proof depends on 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 proof 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 3936  df-iota 5340  df-riota 6041

This theorem is used by:  riotaeqbidv  6044  csbriotag  6055  infvalti  7368  caucvgsrlemfv  8164  axcaucvglemval  8270  axcaucvglemcau  8271  subval  8525  divvalap  9012  divfnzn  10037  flval  10724  cjval  11632  sqrtrval  11788  qnumval  12990  qdenval  12991  grpinvval  13907  uspgredg2v  16667  usgredg2v  16670