Theorem riota2df 5488

Description: A deduction version of riota2f 5489. (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 261	. . 3 |- ( ( ph /\ E! x e. A ps ) -> B e. A )
3		simpr 103	. . . 4 |- ( ( ph /\ E! x e. A ps ) -> E! x e. A ps )
4		df-reu 2313	. . . 4 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
5	3, 4	sylib 127	. . 3 |- ( ( ph /\ E! x e. A ps ) -> E! x ( x e. A /\ ps ) )
6		simpr 103	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x = B )
7	2	adantr 261	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> B e. A )
8	6, 7	eqeltrd 2114	. . . . 5 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x e. A )
9	8	biantrurd 289	. . . 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 446	. . . 4 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ps <-> ch ) )
12	9, 11	bitr3d 179	. . 3 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ( x e. A /\ ps ) <-> ch ) )
13		riota2df.1	. . . 4 |- F/ x ph
14		nfreu1 2481	. . . 4 |- F/ x E! x e. A ps
15	13, 14	nfan 1457	. . 3 |- F/ x ( ph /\ E! x e. A ps )
16		riota2df.3	. . . 4 |- ( ph -> F/ x ch )
17	16	adantr 261	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/ x ch )
18		riota2df.2	. . . 4 |- ( ph -> F/_ x B )
19	18	adantr 261	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/_ x B )
20	2, 5, 12, 15, 17, 19	iota2df 4891	. 2 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota x ( x e. A /\ ps ) ) = B ) )
21		df-riota 5468	. . 3 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
22	21	eqeq1i 2047	. 2 |- ( ( iota_ x e. A ps ) = B <-> ( iota x ( x e. A /\ ps ) ) = B )
23	20, 22	syl6bbr 187	1 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota_ x e. A ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   e. wcel 1393   E!weu 1900   F/_wnfc 2165   E!wreu 2308   iotacio 4865   iota_crio 5467

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867  df-riota 5468

This theorem is referenced by:  riota2f  5489  riota5f  5492