Theorem iota2df 4891

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 103	. . . 4 |- ( ( ph /\ x = B ) -> x = B )
4	3	eqeq2d 2051	. . 3 |- ( ( ph /\ x = B ) -> ( ( iota x ps ) = x <-> ( iota x ps ) = B ) )
5	2, 4	bibi12d 224	. 2 |- ( ( ph /\ x = B ) -> ( ( ps <-> ( iota x ps ) = x ) <-> ( ch <-> ( iota x ps ) = B ) ) )
6		iota2df.2	. . 3 |- ( ph -> E! x ps )
7		iota1 4881	. . 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 4869	. . . . 5 |- F/_ x ( iota x ps )
13	12	a1i 9	. . . 4 |- ( ph -> F/_ x ( iota x ps ) )
14	13, 10	nfeqd 2192	. . 3 |- ( ph -> F/ x ( iota x ps ) = B )
15	11, 14	nfbid 1480	. 2 |- ( ph -> F/ x ( ch <-> ( iota x ps ) = B ) )
16	1, 5, 8, 9, 10, 15	vtocldf 2605	1 |- ( ph -> ( ch <-> ( iota x ps ) = B ) )

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

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-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867

This theorem is referenced by:  iota2d  4892  iota2  4893  riota2df  5488