Theorem nfcsbd 3002

Description: Deduction version of nfcsb 3003. (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 2972	. 2 |- [_ A / y ]_ B = { z | [. A / y ]. z e. B }
2		nfv 1491	. . 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 2269	. . . 4 |- ( ph -> F/ x z e. B )
7	3, 4, 6	nfsbcd 2897	. . 3 |- ( ph -> F/ x [. A / y ]. z e. B )
8	2, 7	nfabd 2274	. 2 |- ( ph -> F/_ x { z | [. A / y ]. z e. B } )
9	1, 8	nfcxfrd 2253	1 |- ( ph -> F/_ x [_ A / y ]_ B )

Syntax hints:   -> wi 4   F/wnf 1419   e. wcel 1463   {cab 2101   F/_wnfc 2242   [.wsbc 2878   [_csb 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-sbc 2879  df-csb 2972

This theorem is referenced by:  nfcsb  3003