Theorem bj-nfs1t

Description: A theorem close to a closed form of nfs1 . (Contributed by BJ, 2-May-2019)

	Ref	Expression
Assertion	bj-nfs1t	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to Ⅎx\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-hbsb3t	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \left(\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi \right)$$
2	1	axc4i	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \forall x\left(\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi \right)$$
3		nf5	$$⊢Ⅎx\left[y/x\right]\phi \leftrightarrow \forall x\left(\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi \right)$$
4	2 3	sylibr	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to Ⅎx\left[y/x\right]\phi$$