Theorem nfraldw

Description: Deduction version of nfralw . Version of nfrald with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 15-Feb-2013) (Revised by Gino Giotto, 24-Sep-2024)

	Ref	Expression
Hypotheses	nfraldw.1	$$⊢Ⅎy\phi$$
	nfraldw.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
	nfraldw.3	$$⊢\phi \to Ⅎx\psi$$
Assertion	nfraldw	$$⊢\phi \to Ⅎx\forall y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		nfraldw.1	$$⊢Ⅎy\phi$$
2		nfraldw.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
3		nfraldw.3	$$⊢\phi \to Ⅎx\psi$$
4		df-ral	$$⊢\forall y\in A\psi \leftrightarrow \forall y\left(y\in A\to \psi \right)$$
5	2	nfcrd	$$⊢\phi \to Ⅎxy\in A$$
6	5 3	nfimd	$$⊢\phi \to Ⅎx\left(y\in A\to \psi \right)$$
7	1 6	nfald	$$⊢\phi \to Ⅎx\forall y\left(y\in A\to \psi \right)$$
8	4 7	nfxfrd	$$⊢\phi \to Ⅎx\forall y\in A\psi$$