Metamath Proof Explorer

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 x A φ x x A φ

Detailed syntax breakdown

Step Hyp Ref Expression
0 vx setvar x
1 cA class A
2 wph wff φ
3 2 0 1 wral wff x A φ
4 0 cv setvar x
5 4 1 wcel wff x A
6 5 2 wi wff x A φ
7 6 0 wal wff x x A φ
8 3 7 wb wff x A φ x x A φ