Theorem wl-dfralfi

Description: Restricted universal quantification ( df-wl-ral ) allows allows a simplification, if we can assume all occurrences of x in A are bounded. (Contributed by Wolf Lammen, 26-May-2023)

	Ref	Expression
Hypothesis	wl-dralfi.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	wl-dfralfi	$$⊢\forall x:A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-dralfi.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		wl-dfralf	$$⊢\underset{\_}{Ⅎ}xA\to \left(\forall x:A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)\right)$$
3	1 2	ax-mp	$$⊢\forall x:A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$