Theorem nfcsbd 3163

Description: Deduction version of nfcsb 3165. (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 3128	. 2 |- [_ A / y ]_ B = { z | [. A / y ]. z e. B }
2		nfv 1576	. . 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 2388	. . . 4 |- ( ph -> F/ x z e. B )
7	3, 4, 6	nfsbcd 3051	. . 3 |- ( ph -> F/ x [. A / y ]. z e. B )
8	2, 7	nfabd 2394	. 2 |- ( ph -> F/_ x { z | [. A / y ]. z e. B } )
9	1, 8	nfcxfrd 2372	1 |- ( ph -> F/_ x [_ A / y ]_ B )

Syntax hints:   -> wi 4   F/wnf 1508   e. wcel 2202   {cab 2217   F/_wnfc 2361   [.wsbc 3031   [_csb 3127

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  df-csb 3128

This theorem is referenced by:  nfcsb  3165