Theorem nfsbcd 3009

Description: Deduction version of nfsbc 3010. (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 2990	. 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 2359	. . 3 |- ( ph -> F/_ x { y | ps } )
6	2, 5	nfeld 2355	. 2 |- ( ph -> F/ x A e. { y | ps } )
7	1, 6	nfxfrd 1489	1 |- ( ph -> F/ x [. A / y ]. ps )

Syntax hints:   -> wi 4   F/wnf 1474   e. wcel 2167   {cab 2182   F/_wnfc 2326   [.wsbc 2989

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-sbc 2990

This theorem is referenced by:  nfsbc  3010  nfcsbd  3120  sbcnestgf  3136