Theorem iota2 4913

Description: The unique element such that ph. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Hypothesis

Ref	Expression
iota2.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
iota2	|- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem iota2

Step	Hyp	Ref	Expression
1		elex 2610	. 2 |- ( A e. B -> A e. _V )
2		simpl 107	. . 3 |- ( ( A e. _V /\ E! x ph ) -> A e. _V )
3		simpr 108	. . 3 |- ( ( A e. _V /\ E! x ph ) -> E! x ph )
4		iota2.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
5	4	adantl 271	. . 3 |- ( ( ( A e. _V /\ E! x ph ) /\ x = A ) -> ( ph <-> ps ) )
6		nfv 1461	. . . 4 |- F/ x A e. _V
7		nfeu1 1952	. . . 4 |- F/ x E! x ph
8	6, 7	nfan 1497	. . 3 |- F/ x ( A e. _V /\ E! x ph )
9		nfvd 1462	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/ x ps )
10		nfcvd 2220	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/_ x A )
11	2, 3, 5, 8, 9, 10	iota2df 4911	. 2 |- ( ( A e. _V /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
12	1, 11	sylan 277	1 |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E!weu 1941   _Vcvv 2601   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)