Theorem iota2df 4911

Description: A condition that allows us to represent "the unique element such that ph " with a class expression A. (Contributed by NM, 30-Dec-2014.)

Hypotheses

Ref	Expression
iota2df.1	|- ( ph -> B e. V )
iota2df.2	|- ( ph -> E! x ps )
iota2df.3	|- ( ( ph /\ x = B ) -> ( ps <-> ch ) )
iota2df.4	|- F/ x ph
iota2df.5	|- ( ph -> F/ x ch )
iota2df.6	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
iota2df	|- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Proof of Theorem iota2df

Step	Hyp	Ref	Expression
1		iota2df.1	. 2 |- ( ph -> B e. V )
2		iota2df.3	. . 3 |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )
3		simpr 108	. . . 4 |- ( ( ph /\ x = B ) -> x = B )
4	3	eqeq2d 2092	. . 3 |- ( ( ph /\ x = B ) -> ( ( iota x ps ) = x <-> ( iota x ps ) = B ) )
5	2, 4	bibi12d 233	. 2 |- ( ( ph /\ x = B ) -> ( ( ps <-> ( iota x ps ) = x ) <-> ( ch <-> ( iota x ps ) = B ) ) )
6		iota2df.2	. . 3 |- ( ph -> E! x ps )
7		iota1 4901	. . 3 |- ( E! x ps -> ( ps <-> ( iota x ps ) = x ) )
8	6, 7	syl 14	. 2 |- ( ph -> ( ps <-> ( iota x ps ) = x ) )
9		iota2df.4	. 2 |- F/ x ph
10		iota2df.6	. 2 |- ( ph -> F/_ x B )
11		iota2df.5	. . 3 |- ( ph -> F/ x ch )
12		nfiota1 4889	. . . . 5 |- F/_ x ( iota x ps )
13	12	a1i 9	. . . 4 |- ( ph -> F/_ x ( iota x ps ) )
14	13, 10	nfeqd 2233	. . 3 |- ( ph -> F/ x ( iota x ps ) = B )
15	11, 14	nfbid 1520	. 2 |- ( ph -> F/ x ( ch <-> ( iota x ps ) = B ) )
16	1, 5, 8, 9, 10, 15	vtocldf 2650	1 |- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   e. wcel 1433   E!weu 1941   F/_wnfc 2206   iotacio 4885

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-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887

This theorem is referenced by:  iota2d  4912  iota2  4913  riota2df  5508