Theorem nfcsbd 2877

Description: Deduction version of nfcsb 2878. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
nfcsbd.1	|- F/yph
nfcsbd.2	|- (ph -> F/_xA)
nfcsbd.3	|- (ph -> F/_xB)

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 2847	. 2 |- [_A / y ]_B = {z | [.A / y ].z e. B }
2		nfv 1418	. . 3 |- F/zph
3		nfcsbd.1	. . . 4 |- F/yph
4		nfcsbd.2	. . . 4 |- (ph -> F/_xA)
5		nfcsbd.3	. . . . 5 |- (ph -> F/_xB)
6	5	nfcrd 2188	. . . 4 |- (ph -> F/x z e. B)
7	3, 4, 6	nfsbcd 2777	. . 3 |- (ph -> F/x [.A / y ].z e. B)
8	2, 7	nfabd 2193	. 2 |- (ph -> F/_x {z | [.A / y ].z e. B })
9	1, 8	nfcxfrd 2173	1 |- (ph -> F/_x [_A / y ]_B)

Syntax hints:   -> wi 4   F/wnf 1346   e. wcel 1390   {cab 2023   F/_wnfc 2162   [.wsbc 2758   [_csb 2846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-sbc 2759  df-csb 2847

This theorem is referenced by:  nfcsb  2878