Theorem nfsbc1d 3064

Description: Deduction version of nfsbc1 3065. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypothesis

Ref	Expression
nfsbc1d.2	|- (ph -> F/_xA)

Assertion

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

Proof of Theorem nfsbc1d

Step	Hyp	Ref	Expression
1		df-sbc 3048	. 2 |- ( [.A / x ].ps <-> A e. {x | ps})
2		nfsbc1d.2	. . 3 |- (ph -> F/_xA)
3		nfab1 2492	. . . 4 |- F/_x{x | ps}
4	3	a1i 10	. . 3 |- (ph -> F/_x{x | ps})
5	2, 4	nfeld 2505	. 2 |- (ph -> F/x A e. {x | ps})
6	1, 5	nfxfrd 1571	1 |- (ph -> F/x [.A / x ].ps)

Syntax hints:   -> wi 4   F/wnf 1544   e. wcel 1710  {cab 2339   F/_wnfc 2477   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sbc 3048

This theorem is referenced by:  nfsbc1  3065  nfcsb1d  3167