Theorem riota1a

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

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

Proof

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