Theorem riota1a 5817

Description: Property of iota. (Contributed by NM, 23-Aug-2011.)

Assertion

Ref	Expression
riota1a	|- ( ( x e. A /\ E! x e. A ph ) -> ( ph <-> ( iota x ( x e. A /\ ph ) ) = x ) )

Proof of Theorem riota1a

Step	Hyp	Ref	Expression
1		ibar 299	. 2 |- ( x e. A -> ( ph <-> ( x e. A /\ ph ) ) )
2		df-reu 2451	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		iota1 5167	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
4	2, 3	sylbi 120	. 2 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
5	1, 4	sylan9bb 458	1 |- ( ( x e. A /\ E! x e. A ph ) -> ( ph <-> ( iota x ( x e. A /\ ph ) ) = x ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E!weu 2014   e. wcel 2136   E!wreu 2446   iotacio 5151

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-reu 2451  df-v 2728  df-sbc 2952  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153

This theorem is referenced by: (None)