Theorem sbcbrg 3942

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 2879	. 2 |- ( y = A -> ( [ y / x ] B R C <-> [. A / x ]. B R C ) )
2		csbeq1 2972	. . 3 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 2972	. . 3 |- ( y = A -> [_ y / x ]_ R = [_ A / x ]_ R )
4		csbeq1 2972	. . 3 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	breq123d 3907	. 2 |- ( y = A -> ( [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
6		nfcsb1v 2999	. . . 4 |- F/_ x [_ y / x ]_ B
7		nfcsb1v 2999	. . . 4 |- F/_ x [_ y / x ]_ R
8		nfcsb1v 2999	. . . 4 |- F/_ x [_ y / x ]_ C
9	6, 7, 8	nfbr 3937	. . 3 |- F/ x [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C
10		csbeq1a 2977	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 2977	. . . 4 |- ( x = y -> R = [_ y / x ]_ R )
12		csbeq1a 2977	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
13	10, 11, 12	breq123d 3907	. . 3 |- ( x = y -> ( B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C ) )
14	9, 13	sbie 1745	. 2 |- ( [ y / x ] B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C )
15	1, 5, 14	vtoclbg 2716	1 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1312   e. wcel 1461   [wsb 1716   [.wsbc 2876   [_csb 2969   class class class wbr 3893

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 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894

This theorem is referenced by:  sbcbr12g  3943  csbcnvg  4681  sbcfung  5103  csbfv12g  5409