Theorem nfsbcd 3051

Description: Deduction version of nfsbc 3052. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
nfsbcd.1	|- F/ y ph
nfsbcd.2	|- ( ph -> F/_ x A )
nfsbcd.3	|- ( ph -> F/ x ps )

Assertion

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

Proof of Theorem nfsbcd

Step	Hyp	Ref	Expression
1		df-sbc 3032	. 2 |- ( [. A / y ]. ps <-> A e. { y | ps } )
2		nfsbcd.2	. . 3 |- ( ph -> F/_ x A )
3		nfsbcd.1	. . . 4 |- F/ y ph
4		nfsbcd.3	. . . 4 |- ( ph -> F/ x ps )
5	3, 4	nfabd 2394	. . 3 |- ( ph -> F/_ x { y | ps } )
6	2, 5	nfeld 2390	. 2 |- ( ph -> F/ x A e. { y | ps } )
7	1, 6	nfxfrd 1523	1 |- ( ph -> F/ x [. A / y ]. ps )

Syntax hints:   -> wi 4   F/wnf 1508   e. wcel 2202   {cab 2217   F/_wnfc 2361   [.wsbc 3031

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-sbc 3032

This theorem is referenced by:  nfsbc  3052  nfcsbd  3163  sbcnestgf  3179