Theorem nfrald

Description: Deduction version of nfral . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfraldw when possible. (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfrald.1	$$⊢Ⅎy\phi$$
2		nfrald.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
3		nfrald.3	$$⊢\phi \to Ⅎx\psi$$
4		df-ral	$$⊢\forall y\in A\psi \leftrightarrow \forall y\left(y\in A\to \psi \right)$$
5		nfcvf	$$⊢\neg \forall xx=y\to \underset{\_}{Ⅎ}xy$$
6	5	adantl	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to \underset{\_}{Ⅎ}xy$$
7	2	adantr	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to \underset{\_}{Ⅎ}xA$$
8	6 7	nfeld	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎxy\in A$$
9	3	adantr	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\psi$$
10	8 9	nfimd	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\left(y\in A\to \psi \right)$$
11	1 10	nfald2	$$⊢\phi \to Ⅎx\forall y\left(y\in A\to \psi \right)$$
12	4 11	nfxfrd	$$⊢\phi \to Ⅎx\forall y\in A\psi$$