Theorem riota1a 5811

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 2449	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		iota1 5161	. . 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 1342   E!weu 2013   e. wcel 2135   E!wreu 2444   iotacio 5145

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-reu 2449  df-v 2723  df-sbc 2947  df-un 3115  df-sn 3576  df-pr 3577  df-uni 3784  df-iota 5147

This theorem is referenced by: (None)