Theorem nfsbc1d 2994

Description: Deduction version of nfsbc1 2995. (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 2978	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
2		nfsbc1d.2	. . 3 |- ( ph -> F/_ x A )
3		nfab1 2334	. . . 4 |- F/_ x { x | ps }
4	3	a1i 9	. . 3 |- ( ph -> F/_ x { x | ps } )
5	2, 4	nfeld 2348	. 2 |- ( ph -> F/ x A e. { x | ps } )
6	1, 5	nfxfrd 1486	1 |- ( ph -> F/ x [. A / x ]. ps )

Syntax hints:   -> wi 4   F/wnf 1471   e. wcel 2160   {cab 2175   F/_wnfc 2319   [.wsbc 2977

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-sbc 2978

This theorem is referenced by:  nfsbc1  2995  nfcsb1d  3103