Theorem riota2df 5508

Description: A deduction version of riota2f 5509. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riota2df.1	|- F/ x ph
riota2df.2	|- ( ph -> F/_ x B )
riota2df.3	|- ( ph -> F/ x ch )
riota2df.4	|- ( ph -> B e. A )
riota2df.5	|- ( ( ph /\ x = B ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
riota2df	|- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota_ x e. A ps ) = B ) )

Distinct variable group:   x, A

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

Proof of Theorem riota2df

Step	Hyp	Ref	Expression
1		riota2df.4	. . . 4 |- ( ph -> B e. A )
2	1	adantr 270	. . 3 |- ( ( ph /\ E! x e. A ps ) -> B e. A )
3		simpr 108	. . . 4 |- ( ( ph /\ E! x e. A ps ) -> E! x e. A ps )
4		df-reu 2355	. . . 4 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
5	3, 4	sylib 120	. . 3 |- ( ( ph /\ E! x e. A ps ) -> E! x ( x e. A /\ ps ) )
6		simpr 108	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x = B )
7	2	adantr 270	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> B e. A )
8	6, 7	eqeltrd 2155	. . . . 5 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x e. A )
9	8	biantrurd 299	. . . 4 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ps <-> ( x e. A /\ ps ) ) )
10		riota2df.5	. . . . 5 |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )
11	10	adantlr 460	. . . 4 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ps <-> ch ) )
12	9, 11	bitr3d 188	. . 3 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ( x e. A /\ ps ) <-> ch ) )
13		riota2df.1	. . . 4 |- F/ x ph
14		nfreu1 2525	. . . 4 |- F/ x E! x e. A ps
15	13, 14	nfan 1497	. . 3 |- F/ x ( ph /\ E! x e. A ps )
16		riota2df.3	. . . 4 |- ( ph -> F/ x ch )
17	16	adantr 270	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/ x ch )
18		riota2df.2	. . . 4 |- ( ph -> F/_ x B )
19	18	adantr 270	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/_ x B )
20	2, 5, 12, 15, 17, 19	iota2df 4911	. 2 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota x ( x e. A /\ ps ) ) = B ) )
21		df-riota 5488	. . 3 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
22	21	eqeq1i 2088	. 2 |- ( ( iota_ x e. A ps ) = B <-> ( iota x ( x e. A /\ ps ) ) = B )
23	20, 22	syl6bbr 196	1 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota_ x e. A ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   e. wcel 1433   E!weu 1941   F/_wnfc 2206   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-3an 921  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:  riota2f  5509  riota5f  5512