Theorem nfsbc1d 2977

Description: Deduction version of nfsbc1 2978. (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 2961	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
2		nfsbc1d.2	. . 3 |- ( ph -> F/_ x A )
3		nfab1 2319	. . . 4 |- F/_ x { x | ps }
4	3	a1i 9	. . 3 |- ( ph -> F/_ x { x | ps } )
5	2, 4	nfeld 2333	. 2 |- ( ph -> F/ x A e. { x | ps } )
6	1, 5	nfxfrd 1473	1 |- ( ph -> F/ x [. A / x ]. ps )

Syntax hints:   -> wi 4   F/wnf 1458   e. wcel 2146   {cab 2161   F/_wnfc 2304   [.wsbc 2960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-sbc 2961

This theorem is referenced by:  nfsbc1  2978  nfcsb1d  3086