Theorem nfcsbd 3116

Description: Deduction version of nfcsb 3118. (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 3081	. 2 |- [_ A / y ]_ B = { z | [. A / y ]. z e. B }
2		nfv 1539	. . 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 2350	. . . 4 |- ( ph -> F/ x z e. B )
7	3, 4, 6	nfsbcd 3005	. . 3 |- ( ph -> F/ x [. A / y ]. z e. B )
8	2, 7	nfabd 2356	. 2 |- ( ph -> F/_ x { z | [. A / y ]. z e. B } )
9	1, 8	nfcxfrd 2334	1 |- ( ph -> F/_ x [_ A / y ]_ B )

Syntax hints:   -> wi 4   F/wnf 1471   e. wcel 2164   {cab 2179   F/_wnfc 2323   [.wsbc 2985   [_csb 3080

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-sbc 2986  df-csb 3081

This theorem is referenced by:  nfcsb  3118