Theorem iota2d 4912

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 ) )

Assertion

Ref	Expression
iota2d	|- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Distinct variable groups:   x, B   ph, x   ch, x

Allowed substitution hints:   ps( x)   V( x)

Proof of Theorem iota2d

Step	Hyp	Ref	Expression
1		iota2df.1	. 2 |- ( ph -> B e. V )
2		iota2df.2	. 2 |- ( ph -> E! x ps )
3		iota2df.3	. 2 |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )
4		nfv 1461	. 2 |- F/ x ph
5		nfvd 1462	. 2 |- ( ph -> F/ x ch )
6		nfcvd 2220	. 2 |- ( ph -> F/_ x B )
7	1, 2, 3, 4, 5, 6	iota2df 4911	1 |- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E!weu 1941   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: (None)