Theorem nfsbc1d 2831

Description: Deduction version of nfsbc1 2832. (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 2816	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
2		nfsbc1d.2	. . 3 |- ( ph -> F/_ x A )
3		nfab1 2221	. . . 4 |- F/_ x { x | ps }
4	3	a1i 9	. . 3 |- ( ph -> F/_ x { x | ps } )
5	2, 4	nfeld 2234	. 2 |- ( ph -> F/ x A e. { x | ps } )
6	1, 5	nfxfrd 1404	1 |- ( ph -> F/ x [. A / x ]. ps )

Syntax hints:   -> wi 4   F/wnf 1389   e. wcel 1433   {cab 2067   F/_wnfc 2206   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-sbc 2816

This theorem is referenced by:  nfsbc1  2832  nfcsb1d  2936