Theorem nfsbc1d 3019

Description: Deduction version of nfsbc1 3020. (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 3003	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
2		nfsbc1d.2	. . 3 |- ( ph -> F/_ x A )
3		nfab1 2351	. . . 4 |- F/_ x { x | ps }
4	3	a1i 9	. . 3 |- ( ph -> F/_ x { x | ps } )
5	2, 4	nfeld 2365	. 2 |- ( ph -> F/ x A e. { x | ps } )
6	1, 5	nfxfrd 1499	1 |- ( ph -> F/ x [. A / x ]. ps )

Syntax hints:   -> wi 4   F/wnf 1484   e. wcel 2177   {cab 2192   F/_wnfc 2336   [.wsbc 3002

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-sbc 3003

This theorem is referenced by:  nfsbc1  3020  nfcsb1d  3128