Theorem riotabidv 5832

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 5199	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. A /\ ch ) ) )
5		df-riota 5830	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 5830	. 2 |- ( iota_ x e. A ch ) = ( iota x ( x e. A /\ ch ) )
7	4, 5, 6	3eqtr4g 2235	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   iotacio 5176   iota_crio 5829

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 3810  df-iota 5178  df-riota 5830

This theorem is referenced by:  riotaeqbidv  5833  csbriotag  5842  infvalti  7020  caucvgsrlemfv  7789  axcaucvglemval  7895  axcaucvglemcau  7896  subval  8148  divvalap  8630  divfnzn  9620  flval  10271  cjval  10853  sqrtrval  11008  qnumval  12184  qdenval  12185  grpinvval  12915