Theorem nfcsbd 2939

Description: Deduction version of nfcsb 2940. (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 2909	. 2 |- [_ A / y ]_ B = { z | [. A / y ]. z e. B }
2		nfv 1461	. . 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 2232	. . . 4 |- ( ph -> F/ x z e. B )
7	3, 4, 6	nfsbcd 2834	. . 3 |- ( ph -> F/ x [. A / y ]. z e. B )
8	2, 7	nfabd 2237	. 2 |- ( ph -> F/_ x { z | [. A / y ]. z e. B } )
9	1, 8	nfcxfrd 2217	1 |- ( ph -> F/_ x [_ A / y ]_ B )

Syntax hints:   -> wi 4   F/wnf 1389   e. wcel 1433   {cab 2067   F/_wnfc 2206   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-sbc 2816  df-csb 2909

This theorem is referenced by:  nfcsb  2940