Theorem ac6c5

Description: Equivalent of Axiom of Choice. B is a collection B ( x ) of nonempty sets. Remark after Theorem 10.46 of TakeutiZaring p. 98. (Contributed by Mario Carneiro, 22-Mar-2013)

	Ref	Expression
Hypotheses	ac6c4.1	|- A e. _V
	ac6c4.2	|- B e. _V
Assertion	ac6c5	|- ( A. x e. A B =/= (/) -> E. f A. x e. A ( f ` x ) e. B )

Proof

Step	Hyp	Ref	Expression
1		ac6c4.1	|- A e. _V
2		ac6c4.2	|- B e. _V
3	1 2	ac6c4	|- ( A. x e. A B =/= (/) -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) )
4		exsimpr	|- ( E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) -> E. f A. x e. A ( f ` x ) e. B )
5	3 4	syl	|- ( A. x e. A B =/= (/) -> E. f A. x e. A ( f ` x ) e. B )