Theorem ac4c

Description: Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997)

	Ref	Expression
Hypothesis	ac4c.1	|- A e. _V
Assertion	ac4c	|- E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x )

Proof

Step	Hyp	Ref	Expression
1		ac4c.1	|- A e. _V
2		raleq	|- ( y = A -> ( A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) <-> A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) )
3	2	exbidv	|- ( y = A -> ( E. f A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) <-> E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) )
4		ac4	|- E. f A. x e. y ( x =/= (/) -> ( f ` x ) e. x )
5	1 3 4	vtocl	|- E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x )