Theorem wl-nfs1t

Description: If y is not free in ph , x is not free in [ y / x ] ph . Closed form of nfs1 . (Contributed by Wolf Lammen, 27-Jul-2019)

	Ref	Expression
Assertion	wl-nfs1t	$$⊢Ⅎy\phi \to Ⅎx\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbequ12r	$$⊢y=x\to \left(\left[y/x\right]\phi \leftrightarrow \phi \right)$$
2	1	equcoms	$$⊢x=y\to \left(\left[y/x\right]\phi \leftrightarrow \phi \right)$$
3	2	sps	$$⊢\forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \phi \right)$$
4	3	drnf1	$$⊢\forall xx=y\to \left(Ⅎx\left[y/x\right]\phi \leftrightarrow Ⅎy\phi \right)$$
5	4	biimprd	$$⊢\forall xx=y\to \left(Ⅎy\phi \to Ⅎx\left[y/x\right]\phi \right)$$
6		nfsb2	$$⊢\neg \forall xx=y\to Ⅎx\left[y/x\right]\phi$$
7	6	a1d	$$⊢\neg \forall xx=y\to \left(Ⅎy\phi \to Ⅎx\left[y/x\right]\phi \right)$$
8	5 7	pm2.61i	$$⊢Ⅎy\phi \to Ⅎx\left[y/x\right]\phi$$