Theorem nfsbv

Description: If z is not free in ph , it is not free in [ y / x ] ph when z is distinct from x and y . Version of nfsb requiring more disjoint variables, but fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016) (Revised by Wolf Lammen, 7-Feb-2023) Remove disjoint variable condition on x , y . (Revised by Steven Nguyen, 13-Aug-2023)

	Ref	Expression
Hypothesis	nfsbv.nf	$$⊢Ⅎz\phi$$
Assertion	nfsbv	$$⊢Ⅎz\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfsbv.nf	$$⊢Ⅎz\phi$$
2		df-sb	$$⊢\left[y/x\right]\phi \leftrightarrow \forall w\left(w=y\to \forall x\left(x=w\to \phi \right)\right)$$
3		nfv	$$⊢Ⅎzw=y$$
4		nfv	$$⊢Ⅎzx=w$$
5	4 1	nfim	$$⊢Ⅎz\left(x=w\to \phi \right)$$
6	5	nfal	$$⊢Ⅎz\forall x\left(x=w\to \phi \right)$$
7	3 6	nfim	$$⊢Ⅎz\left(w=y\to \forall x\left(x=w\to \phi \right)\right)$$
8	7	nfal	$$⊢Ⅎz\forall w\left(w=y\to \forall x\left(x=w\to \phi \right)\right)$$
9	2 8	nfxfr	$$⊢Ⅎz\left[y/x\right]\phi$$