Theorem sbcbrg 4059

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 2967	. 2 |- ( y = A -> ( [ y / x ] B R C <-> [. A / x ]. B R C ) )
2		csbeq1 3062	. . 3 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 3062	. . 3 |- ( y = A -> [_ y / x ]_ R = [_ A / x ]_ R )
4		csbeq1 3062	. . 3 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	breq123d 4019	. 2 |- ( y = A -> ( [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
6		nfcsb1v 3092	. . . 4 |- F/_ x [_ y / x ]_ B
7		nfcsb1v 3092	. . . 4 |- F/_ x [_ y / x ]_ R
8		nfcsb1v 3092	. . . 4 |- F/_ x [_ y / x ]_ C
9	6, 7, 8	nfbr 4051	. . 3 |- F/ x [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C
10		csbeq1a 3068	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 3068	. . . 4 |- ( x = y -> R = [_ y / x ]_ R )
12		csbeq1a 3068	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
13	10, 11, 12	breq123d 4019	. . 3 |- ( x = y -> ( B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C ) )
14	9, 13	sbie 1791	. 2 |- ( [ y / x ] B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C )
15	1, 5, 14	vtoclbg 2800	1 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   [wsb 1762   e. wcel 2148   [.wsbc 2964   [_csb 3059   class class class wbr 4005

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006

This theorem is referenced by:  sbcbr12g  4060  csbcnvg  4813  sbcfung  5242  csbfv12g  5554