Theorem iota2 5280

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 2788	. 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 1552	. . . 4 |- F/ x A e. _V
7		nfeu1 2066	. . . 4 |- F/ x E! x ph
8	6, 7	nfan 1589	. . 3 |- F/ x ( A e. _V /\ E! x ph )
9		nfvd 1553	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/ x ps )
10		nfcvd 2351	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/_ x A )
11	2, 3, 5, 8, 9, 10	iota2df 5276	. 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 1373   E!weu 2055   e. wcel 2178   _Vcvv 2776   iotacio 5249

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-sn 3649  df-pr 3650  df-uni 3865  df-iota 5251

This theorem is referenced by:  iotam  5282  pczpre  12735  pcdiv  12740  gsum0g  13343  gsumval2  13344