Theorem nfsbcdw

Description: Deduction version of nfsbcw . Version of nfsbcd with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 23-Nov-2005) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	nfsbcdw.1	|- F/ y ph
	nfsbcdw.2	|- ( ph -> F/_ x A )
	nfsbcdw.3	|- ( ph -> F/ x ps )
Assertion	nfsbcdw	|- ( ph -> F/ x [. A / y ]. ps )

Proof

Step	Hyp	Ref	Expression
1		nfsbcdw.1	|- F/ y ph
2		nfsbcdw.2	|- ( ph -> F/_ x A )
3		nfsbcdw.3	|- ( ph -> F/ x ps )
4		df-sbc	|- ( [. A / y ]. ps <-> A e. { y | ps } )
5	1 3	nfabdw	|- ( ph -> F/_ x { y | ps } )
6	2 5	nfeld	|- ( ph -> F/ x A e. { y | ps } )
7	4 6	nfxfrd	|- ( ph -> F/ x [. A / y ]. ps )