Theorem sbcbrg 4180

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 3054	. 2 |- ( y = A -> ( [ y / x ] B R C <-> [. A / x ]. B R C ) )
2		csbeq1 3150	. . 3 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
3		csbeq1 3150	. . 3 |- ( y = A -> [_ y / x ]_ R = [_ A / x ]_ R )
4		csbeq1 3150	. . 3 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	breq123d 4139	. 2 |- ( y = A -> ( [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
6		nfcsb1v 3180	. . . 4 |- F/_ x [_ y / x ]_ B
7		nfcsb1v 3180	. . . 4 |- F/_ x [_ y / x ]_ R
8		nfcsb1v 3180	. . . 4 |- F/_ x [_ y / x ]_ C
9	6, 7, 8	nfbr 4172	. . 3 |- F/ x [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C
10		csbeq1a 3156	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
11		csbeq1a 3156	. . . 4 |- ( x = y -> R = [_ y / x ]_ R )
12		csbeq1a 3156	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
13	10, 11, 12	breq123d 4139	. . 3 |- ( x = y -> ( B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C ) )
14	9, 13	sbie 1844	. 2 |- ( [ y / x ] B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C )
15	1, 5, 14	vtoclbg 2884	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 1402   [wsb 1815   e. wcel 2209   [.wsbc 3051   [_csb 3147   class class class wbr 4125

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126

This theorem is referenced by:  sbcbr12g  4181  csbcnvg  4959  sbcfung  5396  csbfv12g  5730