Theorem wl-dfralv

Description: Alternate definition of restricted universal quantification (df-wl-ral ) when x and A are disjoint. (Contributed by Wolf Lammen, 23-May-2023)

	Ref	Expression
Assertion	wl-dfralv	$$⊢\forall x:A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-wl-ral	$$⊢\forall x:A\phi \leftrightarrow \forall y\left(y\in A\to \forall x\left(x=y\to \phi \right)\right)$$
2		19.21v	$$⊢\forall x\left(y\in A\to \left(x=y\to \phi \right)\right)\leftrightarrow \left(y\in A\to \forall x\left(x=y\to \phi \right)\right)$$
3	2	albii	$$⊢\forall y\forall x\left(y\in A\to \left(x=y\to \phi \right)\right)\leftrightarrow \forall y\left(y\in A\to \forall x\left(x=y\to \phi \right)\right)$$
4		wl-dfralflem	$$⊢\forall y\forall x\left(y\in A\to \left(x=y\to \phi \right)\right)\leftrightarrow \forall x\left(x\in A\to \phi \right)$$
5	1 3 4	3bitr2i	$$⊢\forall x:A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$