Theorem riota1a 5897

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 301	. 2 |- ( x e. A -> ( ph <-> ( x e. A /\ ph ) ) )
2		df-reu 2482	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		iota1 5233	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
4	2, 3	sylbi 121	. 2 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
5	1, 4	sylan9bb 462	1 |- ( ( x e. A /\ E! x e. A ph ) -> ( ph <-> ( iota x ( x e. A /\ ph ) ) = x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E!weu 2045   e. wcel 2167   E!wreu 2477   iotacio 5217

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-reu 2482  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219

This theorem is referenced by: (None)