Theorem riotaprop 6188

Description: Properties of a restricted definite description operator. Todo (df-riota 6164 update): can some uses of riota2f 6186 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 6179	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
3	1, 2	syl5eqel 2546	. 2 |- ( E! x e. A ph -> B e. A )
4	1	eqcomi 2467	. . . 4 |- ( iota_ x e. A ph ) = B
5		nfriota1 6171	. . . . . 6 |- F/_ x ( iota_ x e. A ph )
6	1, 5	nfcxfr 2614	. . . . 5 |- F/_ x B
7		riotaprop.0	. . . . 5 |- F/ x ps
8		riotaprop.2	. . . . 5 |- ( x = B -> ( ph <-> ps ) )
9	6, 7, 8	riota2f 6186	. . . 4 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
10	4, 9	mpbiri 233	. . 3 |- ( ( B e. A /\ E! x e. A ph ) -> ps )
11	3, 10	mpancom 669	. 2 |- ( E! x e. A ph -> ps )
12	3, 11	jca 532	1 |- ( E! x e. A ph -> ( B e. A /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   F/wnf 1590   e. wcel 1758   E!wreu 2801   iota_crio 6163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-un 3444  df-in 3446  df-ss 3453  df-sn 3989  df-pr 3991  df-uni 4203  df-iota 5492  df-riota 6164

This theorem is referenced by:  fin23lem27  8611  lble  10396  ltrniotaval  34583