Theorem riotaxfrd 6642

Description: Change the variable x in the expression for "the unique x such that ps " to another variable y contained in expression B. Use reuhypd 4895 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riotaxfrd.1	|- F/_ y C
riotaxfrd.2	|- ( ( ph /\ y e. A ) -> B e. A )
riotaxfrd.3	|- ( ( ph /\ ( iota_ y e. A ch ) e. A ) -> C e. A )
riotaxfrd.4	|- ( x = B -> ( ps <-> ch ) )
riotaxfrd.5	|- ( y = ( iota_ y e. A ch ) -> B = C )
riotaxfrd.6	|- ( ( ph /\ x e. A ) -> E! y e. A x = B )

Assertion

Ref	Expression
riotaxfrd	|- ( ( ph /\ E! x e. A ps ) -> ( iota_ x e. A ps ) = C )

Distinct variable groups:   x, B   x, C   x, y, A   ph, x, y   ps, y   ch, x

Allowed substitution hints:   ps( x)   ch( y)   B( y)   C( y)

Proof of Theorem riotaxfrd

Step	Hyp	Ref	Expression
1		rabid 3116	. . . 4 |- ( x e. { x e. A | ps } <-> ( x e. A /\ ps ) )
2	1	baib 944	. . 3 |- ( x e. A -> ( x e. { x e. A | ps } <-> ps ) )
3	2	riotabiia 6628	. 2 |- ( iota_ x e. A x e. { x e. A | ps } ) = ( iota_ x e. A ps )
4		riotaxfrd.2	. . . . . 6 |- ( ( ph /\ y e. A ) -> B e. A )
5		riotaxfrd.6	. . . . . 6 |- ( ( ph /\ x e. A ) -> E! y e. A x = B )
6		riotaxfrd.4	. . . . . 6 |- ( x = B -> ( ps <-> ch ) )
7	4, 5, 6	reuxfrd 4893	. . . . 5 |- ( ph -> ( E! x e. A ps <-> E! y e. A ch ) )
8		riotacl2 6624	. . . . . . . 8 |- ( E! y e. A ch -> ( iota_ y e. A ch ) e. { y e. A | ch } )
9	8	adantl 482	. . . . . . 7 |- ( ( ph /\ E! y e. A ch ) -> ( iota_ y e. A ch ) e. { y e. A | ch } )
10		riotacl 6625	. . . . . . . 8 |- ( E! y e. A ch -> ( iota_ y e. A ch ) e. A )
11		nfriota1 6618	. . . . . . . . 9 |- F/_ y ( iota_ y e. A ch )
12		riotaxfrd.1	. . . . . . . . 9 |- F/_ y C
13		riotaxfrd.5	. . . . . . . . 9 |- ( y = ( iota_ y e. A ch ) -> B = C )
14	11, 12, 4, 6, 13	rabxfrd 4889	. . . . . . . 8 |- ( ( ph /\ ( iota_ y e. A ch ) e. A ) -> ( C e. { x e. A | ps } <-> ( iota_ y e. A ch ) e. { y e. A | ch } ) )
15	10, 14	sylan2 491	. . . . . . 7 |- ( ( ph /\ E! y e. A ch ) -> ( C e. { x e. A | ps } <-> ( iota_ y e. A ch ) e. { y e. A | ch } ) )
16	9, 15	mpbird 247	. . . . . 6 |- ( ( ph /\ E! y e. A ch ) -> C e. { x e. A | ps } )
17	16	ex 450	. . . . 5 |- ( ph -> ( E! y e. A ch -> C e. { x e. A | ps } ) )
18	7, 17	sylbid 230	. . . 4 |- ( ph -> ( E! x e. A ps -> C e. { x e. A | ps } ) )
19	18	imp 445	. . 3 |- ( ( ph /\ E! x e. A ps ) -> C e. { x e. A | ps } )
20		riotaxfrd.3	. . . . . . . 8 |- ( ( ph /\ ( iota_ y e. A ch ) e. A ) -> C e. A )
21	20	ex 450	. . . . . . 7 |- ( ph -> ( ( iota_ y e. A ch ) e. A -> C e. A ) )
22	10, 21	syl5 34	. . . . . 6 |- ( ph -> ( E! y e. A ch -> C e. A ) )
23	7, 22	sylbid 230	. . . . 5 |- ( ph -> ( E! x e. A ps -> C e. A ) )
24	23	imp 445	. . . 4 |- ( ( ph /\ E! x e. A ps ) -> C e. A )
25	1	baibr 945	. . . . . . 7 |- ( x e. A -> ( ps <-> x e. { x e. A | ps } ) )
26	25	reubiia 3127	. . . . . 6 |- ( E! x e. A ps <-> E! x e. A x e. { x e. A | ps } )
27	26	biimpi 206	. . . . 5 |- ( E! x e. A ps -> E! x e. A x e. { x e. A | ps } )
28	27	adantl 482	. . . 4 |- ( ( ph /\ E! x e. A ps ) -> E! x e. A x e. { x e. A | ps } )
29		nfcv 2764	. . . . 5 |- F/_ x C
30		nfrab1 3122	. . . . . 6 |- F/_ x { x e. A | ps }
31	30	nfel2 2781	. . . . 5 |- F/ x C e. { x e. A | ps }
32		eleq1 2689	. . . . 5 |- ( x = C -> ( x e. { x e. A | ps } <-> C e. { x e. A | ps } ) )
33	29, 31, 32	riota2f 6632	. . . 4 |- ( ( C e. A /\ E! x e. A x e. { x e. A | ps } ) -> ( C e. { x e. A | ps } <-> ( iota_ x e. A x e. { x e. A | ps } ) = C ) )
34	24, 28, 33	syl2anc 693	. . 3 |- ( ( ph /\ E! x e. A ps ) -> ( C e. { x e. A | ps } <-> ( iota_ x e. A x e. { x e. A | ps } ) = C ) )
35	19, 34	mpbid 222	. 2 |- ( ( ph /\ E! x e. A ps ) -> ( iota_ x e. A x e. { x e. A | ps } ) = C )
36	3, 35	syl5eqr 2670	1 |- ( ( ph /\ E! x e. A ps ) -> ( iota_ x e. A ps ) = C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   E!wreu 2914   {crab 2916   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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-riota 6611

This theorem is referenced by:  riotaneg  11002  zriotaneg  11491  riotaocN  34496