Theorem nfsbcd 2982

Description: Deduction version of nfsbc 2983. (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 2963	. 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 2339	. . 3 |- ( ph -> F/_ x { y | ps } )
6	2, 5	nfeld 2335	. 2 |- ( ph -> F/ x A e. { y | ps } )
7	1, 6	nfxfrd 1475	1 |- ( ph -> F/ x [. A / y ]. ps )

Syntax hints:   -> wi 4   F/wnf 1460   e. wcel 2148   {cab 2163   F/_wnfc 2306   [.wsbc 2962

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-sbc 2963

This theorem is referenced by:  nfsbc  2983  nfcsbd  3092  sbcnestgf  3108