Theorem nfsbcd 2881

Description: Deduction version of nfsbc 2882. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfsbcd	|- ( ph -> F/ x [. A / y ]. ps )

Proof of Theorem nfsbcd

Step	Hyp	Ref	Expression
1		df-sbc 2863	. 2 |- ( [. A / y ]. ps <-> A e. { y | ps } )
2		nfsbcd.2	. . 3 |- ( ph -> F/_ x A )
3		nfsbcd.1	. . . 4 |- F/ y ph
4		nfsbcd.3	. . . 4 |- ( ph -> F/ x ps )
5	3, 4	nfabd 2259	. . 3 |- ( ph -> F/_ x { y | ps } )
6	2, 5	nfeld 2256	. 2 |- ( ph -> F/ x A e. { y | ps } )
7	1, 6	nfxfrd 1419	1 |- ( ph -> F/ x [. A / y ]. ps )

Syntax hints:   -> wi 4   F/wnf 1404   e. wcel 1448   {cab 2086   F/_wnfc 2227   [.wsbc 2862

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-sbc 2863

This theorem is referenced by:  nfsbc  2882  nfcsbd  2986  sbcnestgf  3001