Theorem riota2df 5829

Description: A deduction version of riota2f 5830. (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 274	. . 3 |- ( ( ph /\ E! x e. A ps ) -> B e. A )
3		simpr 109	. . . 4 |- ( ( ph /\ E! x e. A ps ) -> E! x e. A ps )
4		df-reu 2455	. . . 4 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
5	3, 4	sylib 121	. . 3 |- ( ( ph /\ E! x e. A ps ) -> E! x ( x e. A /\ ps ) )
6		simpr 109	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x = B )
7	2	adantr 274	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> B e. A )
8	6, 7	eqeltrd 2247	. . . . 5 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x e. A )
9	8	biantrurd 303	. . . 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 474	. . . 4 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ps <-> ch ) )
12	9, 11	bitr3d 189	. . 3 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ( x e. A /\ ps ) <-> ch ) )
13		riota2df.1	. . . 4 |- F/ x ph
14		nfreu1 2641	. . . 4 |- F/ x E! x e. A ps
15	13, 14	nfan 1558	. . 3 |- F/ x ( ph /\ E! x e. A ps )
16		riota2df.3	. . . 4 |- ( ph -> F/ x ch )
17	16	adantr 274	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/ x ch )
18		riota2df.2	. . . 4 |- ( ph -> F/_ x B )
19	18	adantr 274	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/_ x B )
20	2, 5, 12, 15, 17, 19	iota2df 5184	. 2 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota x ( x e. A /\ ps ) ) = B ) )
21		df-riota 5809	. . 3 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
22	21	eqeq1i 2178	. 2 |- ( ( iota_ x e. A ps ) = B <-> ( iota x ( x e. A /\ ps ) ) = B )
23	20, 22	bitr4di 197	1 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota_ x e. A ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   F/wnf 1453   E!weu 2019   e. wcel 2141   F/_wnfc 2299   E!wreu 2450   iotacio 5158   iota_crio 5808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160  df-riota 5809

This theorem is referenced by:  riota2f  5830  riota5f  5833