Theorem iota2df 5240

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 110	. . . 4 |- ( ( ph /\ x = B ) -> x = B )
4	3	eqeq2d 2205	. . 3 |- ( ( ph /\ x = B ) -> ( ( iota x ps ) = x <-> ( iota x ps ) = B ) )
5	2, 4	bibi12d 235	. 2 |- ( ( ph /\ x = B ) -> ( ( ps <-> ( iota x ps ) = x ) <-> ( ch <-> ( iota x ps ) = B ) ) )
6		iota2df.2	. . 3 |- ( ph -> E! x ps )
7		iota1 5229	. . 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 5217	. . . . 5 |- F/_ x ( iota x ps )
13	12	a1i 9	. . . 4 |- ( ph -> F/_ x ( iota x ps ) )
14	13, 10	nfeqd 2351	. . 3 |- ( ph -> F/ x ( iota x ps ) = B )
15	11, 14	nfbid 1599	. 2 |- ( ph -> F/ x ( ch <-> ( iota x ps ) = B ) )
16	1, 5, 8, 9, 10, 15	vtocldf 2811	1 |- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1471   E!weu 2042   e. wcel 2164   F/_wnfc 2323   iotacio 5213

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-sn 3624  df-pr 3625  df-uni 3836  df-iota 5215

This theorem is referenced by:  iota2d  5241  iota2  5244  riota2df  5894