Theorem riotaprop 6290

Description: Properties of a restricted definite description operator. Todo (df-riota 6267 update): can some uses of riota2f 6288 be shortened with this? (Contributed by NM, 23-Nov-2013.)

Hypotheses

Ref	Expression
riotaprop.0	|- F/ x ps
riotaprop.1	|- B = ( iota_ x e. A ph )
riotaprop.2	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
riotaprop	|- ( E! x e. A ph -> ( B e. A /\ ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   B( x)

Proof of Theorem riotaprop

Step	Hyp	Ref	Expression
1		riotaprop.1	. . 3 |- B = ( iota_ x e. A ph )
2		riotacl 6281	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
3	1, 2	syl5eqel 2511	. 2 |- ( E! x e. A ph -> B e. A )
4	1	eqcomi 2435	. . . 4 |- ( iota_ x e. A ph ) = B
5		nfriota1 6274	. . . . . 6 |- F/_ x ( iota_ x e. A ph )
6	1, 5	nfcxfr 2578	. . . . 5 |- F/_ x B
7		riotaprop.0	. . . . 5 |- F/ x ps
8		riotaprop.2	. . . . 5 |- ( x = B -> ( ph <-> ps ) )
9	6, 7, 8	riota2f 6288	. . . 4 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
10	4, 9	mpbiri 236	. . 3 |- ( ( B e. A /\ E! x e. A ph ) -> ps )
11	3, 10	mpancom 673	. 2 |- ( E! x e. A ph -> ps )
12	3, 11	jca 534	1 |- ( E! x e. A ph -> ( B e. A /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   F/wnf 1661   e. wcel 1872   E!wreu 2773   iota_crio 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-un 3441  df-in 3443  df-ss 3450  df-sn 3999  df-pr 4001  df-uni 4220  df-iota 5565  df-riota 6267

This theorem is referenced by:  fin23lem27  8765  lble  10565  ltrniotaval  34117