Theorem sbcbrg 3834

Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
sbcbrg	|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )

Proof of Theorem sbcbrg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2818	. 2 |- ( y = A -> ( [ y / x ] B R C <-> [. A / x ]. B R C ) )
2		csbeq1 2911	. . 3 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 2911	. . 3 |- ( y = A -> [_ y / x ]_ R = [_ A / x ]_ R )
4		csbeq1 2911	. . 3 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	breq123d 3799	. 2 |- ( y = A -> ( [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
6		nfcsb1v 2938	. . . 4 |- F/_ x [_ y / x ]_ B
7		nfcsb1v 2938	. . . 4 |- F/_ x [_ y / x ]_ R
8		nfcsb1v 2938	. . . 4 |- F/_ x [_ y / x ]_ C
9	6, 7, 8	nfbr 3829	. . 3 |- F/ x [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C
10		csbeq1a 2916	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 2916	. . . 4 |- ( x = y -> R = [_ y / x ]_ R )
12		csbeq1a 2916	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
13	10, 11, 12	breq123d 3799	. . 3 |- ( x = y -> ( B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C ) )
14	9, 13	sbie 1714	. 2 |- ( [ y / x ] B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C )
15	1, 5, 14	vtoclbg 2659	1 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   [wsb 1685   [.wsbc 2815   [_csb 2908   class class class wbr 3785

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-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786

This theorem is referenced by:  sbcbr12g  3835  csbcnvg  4537  sbcfung  4945  csbfv12g  5230