Theorem riotaclb 6649

Description: Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem riotaclb

Step	Hyp	Ref	Expression
1		riotacl 6625	. 2 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
2		riotaund 6647	. . . . . 6 |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )
3	2	eleq1d 2686	. . . . 5 |- ( -. E! x e. A ph -> ( ( iota_ x e. A ph ) e. A <-> (/) e. A ) )
4	3	notbid 308	. . . 4 |- ( -. E! x e. A ph -> ( -. ( iota_ x e. A ph ) e. A <-> -. (/) e. A ) )
5	4	biimprcd 240	. . 3 |- ( -. (/) e. A -> ( -. E! x e. A ph -> -. ( iota_ x e. A ph ) e. A ) )
6	5	con4d 114	. 2 |- ( -. (/) e. A -> ( ( iota_ x e. A ph ) e. A -> E! x e. A ph ) )
7	1, 6	impbid2 216	1 |- ( -. (/) e. A -> ( E! x e. A ph <-> ( iota_ x e. A ph ) e. A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   e. wcel 1990   E!wreu 2914   (/)c0 3915   iota_crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-riota 6611

This theorem is referenced by: (None)