Theorem sbcbrg 4143

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 3034	. 2 |- ( y = A -> ( [ y / x ] B R C <-> [. A / x ]. B R C ) )
2		csbeq1 3130	. . 3 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 3130	. . 3 |- ( y = A -> [_ y / x ]_ R = [_ A / x ]_ R )
4		csbeq1 3130	. . 3 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	breq123d 4102	. 2 |- ( y = A -> ( [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
6		nfcsb1v 3160	. . . 4 |- F/_ x [_ y / x ]_ B
7		nfcsb1v 3160	. . . 4 |- F/_ x [_ y / x ]_ R
8		nfcsb1v 3160	. . . 4 |- F/_ x [_ y / x ]_ C
9	6, 7, 8	nfbr 4135	. . 3 |- F/ x [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C
10		csbeq1a 3136	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 3136	. . . 4 |- ( x = y -> R = [_ y / x ]_ R )
12		csbeq1a 3136	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
13	10, 11, 12	breq123d 4102	. . 3 |- ( x = y -> ( B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C ) )
14	9, 13	sbie 1839	. 2 |- ( [ y / x ] B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C )
15	1, 5, 14	vtoclbg 2865	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 1397   [wsb 1810   e. wcel 2202   [.wsbc 3031   [_csb 3127   class class class wbr 4088

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  sbcbr12g  4144  csbcnvg  4914  sbcfung  5350  csbfv12g  5679