Theorem riota2df 6631

Description: A deduction version of riota2f 6632. (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 481	. . 3 |- ( ( ph /\ E! x e. A ps ) -> B e. A )
3		simpr 477	. . . 4 |- ( ( ph /\ E! x e. A ps ) -> E! x e. A ps )
4		df-reu 2919	. . . 4 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
5	3, 4	sylib 208	. . 3 |- ( ( ph /\ E! x e. A ps ) -> E! x ( x e. A /\ ps ) )
6		simpr 477	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x = B )
7	2	adantr 481	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> B e. A )
8	6, 7	eqeltrd 2701	. . . . 5 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x e. A )
9	8	biantrurd 529	. . . 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 751	. . . 4 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ps <-> ch ) )
12	9, 11	bitr3d 270	. . 3 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ( x e. A /\ ps ) <-> ch ) )
13		riota2df.1	. . . 4 |- F/ x ph
14		nfreu1 3110	. . . 4 |- F/ x E! x e. A ps
15	13, 14	nfan 1828	. . 3 |- F/ x ( ph /\ E! x e. A ps )
16		riota2df.3	. . . 4 |- ( ph -> F/ x ch )
17	16	adantr 481	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/ x ch )
18		riota2df.2	. . . 4 |- ( ph -> F/_ x B )
19	18	adantr 481	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/_ x B )
20	2, 5, 12, 15, 17, 19	iota2df 5875	. 2 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota x ( x e. A /\ ps ) ) = B ) )
21		df-riota 6611	. . 3 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
22	21	eqeq1i 2627	. 2 |- ( ( iota_ x e. A ps ) = B <-> ( iota x ( x e. A /\ ps ) ) = B )
23	20, 22	syl6bbr 278	1 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota_ x e. A ps ) = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   E!weu 2470   F/_wnfc 2751   E!wreu 2914   iotacio 5849   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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-riota 6611

This theorem is referenced by:  riota2f  6632  riotaeqimp  6634  riota5f  6636  mapdheq  37017  hdmap1eq  37091  hdmapval2lem  37123