Definition df-ral

Description: Define restricted universal quantification. Special case of Definition 4.15(3) of TakeutiZaring p. 22.

Note: This notation is most often used to express that ph holds for all elements of a given class A . For this reading F/_ x A is required, though, for example, asserted when x and A are disjoint.

Should instead A depend on x , you rather focus on those x that happen to be contained in the corresponding A ( x ) . This hardly used interpretation could still occur naturally. For some examples, look at ralndv1 or ralndv2 , courtesy of AV.

So be careful to either keep A independent of x , or adjust your comments to include such exotic cases. (Contributed by NM, 19-Aug-1993)

	Ref	Expression
Assertion	df-ral	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		vx	|- x
1		cA	|- A
2		wph	|- ph
3	2 0 1	wral	|- A. x e. A ph
4	0	cv	|- x
5	4 1	wcel	|- x e. A
6	5 2	wi	|- ( x e. A -> ph )
7	6 0	wal	|- A. x ( x e. A -> ph )
8	3 7	wb	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )