Theorem nfsbcd

Description: Deduction version of nfsbc . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfsbcdw when possible. (Contributed by NM, 23-Nov-2005) (Revised by Mario Carneiro, 12-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfsbcd.1	|- F/ y ph
2		nfsbcd.2	|- ( ph -> F/_ x A )
3		nfsbcd.3	|- ( ph -> F/ x ps )
4		df-sbc	|- ( [. A / y ]. ps <-> A e. { y | ps } )
5	1 3	nfabd	|- ( 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 )