# 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 ${⊢}\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\to {\phi }\right)$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 vx ${setvar}{x}$
1 cA ${class}{A}$
2 wph ${wff}{\phi }$
3 2 0 1 wral ${wff}\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }$
4 0 cv ${setvar}{x}$
5 4 1 wcel ${wff}{x}\in {A}$
6 5 2 wi ${wff}\left({x}\in {A}\to {\phi }\right)$
7 6 0 wal ${wff}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\to {\phi }\right)$
8 3 7 wb ${wff}\left(\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}\in {A}\to {\phi }\right)\right)$