Theorem riota2df 5854

Description: A deduction version of riota2f 5855. (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 276	. . 3 |- ( ( ph /\ E! x e. A ps ) -> B e. A )
3		simpr 110	. . . 4 |- ( ( ph /\ E! x e. A ps ) -> E! x e. A ps )
4		df-reu 2462	. . . 4 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
5	3, 4	sylib 122	. . 3 |- ( ( ph /\ E! x e. A ps ) -> E! x ( x e. A /\ ps ) )
6		simpr 110	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x = B )
7	2	adantr 276	. . . . . 6 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> B e. A )
8	6, 7	eqeltrd 2254	. . . . 5 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> x e. A )
9	8	biantrurd 305	. . . 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 477	. . . 4 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ps <-> ch ) )
12	9, 11	bitr3d 190	. . 3 |- ( ( ( ph /\ E! x e. A ps ) /\ x = B ) -> ( ( x e. A /\ ps ) <-> ch ) )
13		riota2df.1	. . . 4 |- F/ x ph
14		nfreu1 2649	. . . 4 |- F/ x E! x e. A ps
15	13, 14	nfan 1565	. . 3 |- F/ x ( ph /\ E! x e. A ps )
16		riota2df.3	. . . 4 |- ( ph -> F/ x ch )
17	16	adantr 276	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/ x ch )
18		riota2df.2	. . . 4 |- ( ph -> F/_ x B )
19	18	adantr 276	. . 3 |- ( ( ph /\ E! x e. A ps ) -> F/_ x B )
20	2, 5, 12, 15, 17, 19	iota2df 5204	. 2 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota x ( x e. A /\ ps ) ) = B ) )
21		df-riota 5834	. . 3 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
22	21	eqeq1i 2185	. 2 |- ( ( iota_ x e. A ps ) = B <-> ( iota x ( x e. A /\ ps ) ) = B )
23	20, 22	bitr4di 198	1 |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota_ x e. A ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   F/wnf 1460   E!weu 2026   e. wcel 2148   F/_wnfc 2306   E!wreu 2457   iotacio 5178   iota_crio 5833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-v 2741  df-sbc 2965  df-un 3135  df-sn 3600  df-pr 3601  df-uni 3812  df-iota 5180  df-riota 5834

This theorem is referenced by:  riota2f  5855  riota5f  5858