Theorem iota2 5308

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 2811	. 2 |- ( A e. B -> A e. _V )
2		simpl 109	. . 3 |- ( ( A e. _V /\ E! x ph ) -> A e. _V )
3		simpr 110	. . 3 |- ( ( A e. _V /\ E! x ph ) -> E! x ph )
4		iota2.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
5	4	adantl 277	. . 3 |- ( ( ( A e. _V /\ E! x ph ) /\ x = A ) -> ( ph <-> ps ) )
6		nfv 1574	. . . 4 |- F/ x A e. _V
7		nfeu1 2088	. . . 4 |- F/ x E! x ph
8	6, 7	nfan 1611	. . 3 |- F/ x ( A e. _V /\ E! x ph )
9		nfvd 1575	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/ x ps )
10		nfcvd 2373	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/_ x A )
11	2, 3, 5, 8, 9, 10	iota2df 5304	. 2 |- ( ( A e. _V /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
12	1, 11	sylan 283	1 |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E!weu 2077   e. wcel 2200   _Vcvv 2799   iotacio 5276

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3889  df-iota 5278

This theorem is referenced by:  iotam  5310  fvmbr  5662  pczpre  12820  pcdiv  12825  gsum0g  13429  gsumval2  13430