Theorem iota2 5109

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 2692	. 2 |- ( A e. B -> A e. _V )
2		simpl 108	. . 3 |- ( ( A e. _V /\ E! x ph ) -> A e. _V )
3		simpr 109	. . 3 |- ( ( A e. _V /\ E! x ph ) -> E! x ph )
4		iota2.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
5	4	adantl 275	. . 3 |- ( ( ( A e. _V /\ E! x ph ) /\ x = A ) -> ( ph <-> ps ) )
6		nfv 1508	. . . 4 |- F/ x A e. _V
7		nfeu1 2008	. . . 4 |- F/ x E! x ph
8	6, 7	nfan 1544	. . 3 |- F/ x ( A e. _V /\ E! x ph )
9		nfvd 1509	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/ x ps )
10		nfcvd 2280	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/_ x A )
11	2, 3, 5, 8, 9, 10	iota2df 5107	. 2 |- ( ( A e. _V /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
12	1, 11	sylan 281	1 |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E!weu 1997   _Vcvv 2681   iotacio 5081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-sn 3528  df-pr 3529  df-uni 3732  df-iota 5083

This theorem is referenced by: (None)