Theorem riota1 5506

Description: Property of restricted iota. Compare iota1 4901. (Contributed by Mario Carneiro, 15-Oct-2016.)

Assertion

Ref	Expression
riota1	|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem riota1

Step	Hyp	Ref	Expression
1		df-reu 2355	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iota1 4901	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
3	1, 2	sylbi 119	. 2 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
4		df-riota 5488	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5	4	eqeq1i 2088	. 2 |- ( ( iota_ x e. A ph ) = x <-> ( iota x ( x e. A /\ ph ) ) = x )
6	3, 5	syl6bbr 196	1 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E!weu 1941   E!wreu 2350   iotacio 4885   iota_crio 5487

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-reu 2355  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by:  supelti  6415  oddpwdclemdvds  10548  oddpwdclemndvds  10549