Theorem nfcsbd 3080

Description: Deduction version of nfcsb 3082. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfcsbd	|- ( ph -> F/_ x [_ A / y ]_ B )

Proof of Theorem nfcsbd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3046	. 2 |- [_ A / y ]_ B = { z | [. A / y ]. z e. B }
2		nfv 1516	. . 3 |- F/ z ph
3		nfcsbd.1	. . . 4 |- F/ y ph
4		nfcsbd.2	. . . 4 |- ( ph -> F/_ x A )
5		nfcsbd.3	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2322	. . . 4 |- ( ph -> F/ x z e. B )
7	3, 4, 6	nfsbcd 2970	. . 3 |- ( ph -> F/ x [. A / y ]. z e. B )
8	2, 7	nfabd 2328	. 2 |- ( ph -> F/_ x { z | [. A / y ]. z e. B } )
9	1, 8	nfcxfrd 2306	1 |- ( ph -> F/_ x [_ A / y ]_ B )

Syntax hints:   -> wi 4   F/wnf 1448   e. wcel 2136   {cab 2151   F/_wnfc 2295   [.wsbc 2951   [_csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-sbc 2952  df-csb 3046

This theorem is referenced by:  nfcsb  3082