Theorem iota2df 5244

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 2208	. . 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 5233	. . 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 5221	. . . . 5 |- F/_ x ( iota x ps )
13	12	a1i 9	. . . 4 |- ( ph -> F/_ x ( iota x ps ) )
14	13, 10	nfeqd 2354	. . 3 |- ( ph -> F/ x ( iota x ps ) = B )
15	11, 14	nfbid 1602	. 2 |- ( ph -> F/ x ( ch <-> ( iota x ps ) = B ) )
16	1, 5, 8, 9, 10, 15	vtocldf 2815	1 |- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1474   E!weu 2045   e. wcel 2167   F/_wnfc 2326   iotacio 5217

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219

This theorem is referenced by:  iota2d  5245  iota2  5248  riota2df  5898