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	|- F/ z ph
Assertion	nfsbv	|- F/ z [ y / x ] ph

Proof

Step	Hyp	Ref	Expression
1		nfsbv.nf	|- F/ z ph
2		df-sb	|- ( [ y / x ] ph <-> A. w ( w = y -> A. x ( x = w -> ph ) ) )
3		nfv	|- F/ z w = y
4		nfv	|- F/ z x = w
5	4 1	nfim	|- F/ z ( x = w -> ph )
6	5	nfal	|- F/ z A. x ( x = w -> ph )
7	3 6	nfim	|- F/ z ( w = y -> A. x ( x = w -> ph ) )
8	7	nfal	|- F/ z A. w ( w = y -> A. x ( x = w -> ph ) )
9	2 8	nfxfr	|- F/ z [ y / x ] ph