Theorem nfsbc1d 2967

Description: Deduction version of nfsbc1 2968. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypothesis

Ref	Expression
nfsbc1d.2	|- ( ph -> F/_ x A )

Assertion

Ref	Expression
nfsbc1d	|- ( ph -> F/ x [. A / x ]. ps )

Proof of Theorem nfsbc1d

Step	Hyp	Ref	Expression
1		df-sbc 2952	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
2		nfsbc1d.2	. . 3 |- ( ph -> F/_ x A )
3		nfab1 2310	. . . 4 |- F/_ x { x | ps }
4	3	a1i 9	. . 3 |- ( ph -> F/_ x { x | ps } )
5	2, 4	nfeld 2324	. 2 |- ( ph -> F/ x A e. { x | ps } )
6	1, 5	nfxfrd 1463	1 |- ( ph -> F/ x [. A / x ]. ps )

Syntax hints:   -> wi 4   F/wnf 1448   e. wcel 2136   {cab 2151   F/_wnfc 2295   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-sbc 2952

This theorem is referenced by:  nfsbc1  2968  nfcsb1d  3076