Theorem sbcbr123 4706

Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Modified by NM, 22-Aug-2018.)

Assertion

Ref	Expression
sbcbr123	|- ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C )

Proof of Theorem sbcbr123

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3445	. 2 |- ( [. A / x ]. B R C -> A e. _V )
2		br0 4701	. . . 4 |- -. [_ A / x ]_ B (/) [_ A / x ]_ C
3		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ R = (/) )
4	3	breqd 4664	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C <-> [_ A / x ]_ B (/) [_ A / x ]_ C ) )
5	2, 4	mtbiri 317	. . 3 |- ( -. A e. _V -> -. [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C )
6	5	con4i 113	. 2 |- ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C -> A e. _V )
7		dfsbcq2 3438	. . 3 |- ( y = A -> ( [ y / x ] B R C <-> [. A / x ]. B R C ) )
8		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
9		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ R = [_ A / x ]_ R )
10		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
11	8, 9, 10	breq123d 4667	. . 3 |- ( y = A -> ( [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
12		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ B
13		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ R
14		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ C
15	12, 13, 14	nfbr 4699	. . . 4 |- F/ x [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C
16		csbeq1a 3542	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
17		csbeq1a 3542	. . . . 5 |- ( x = y -> R = [_ y / x ]_ R )
18		csbeq1a 3542	. . . . 5 |- ( x = y -> C = [_ y / x ]_ C )
19	16, 17, 18	breq123d 4667	. . . 4 |- ( x = y -> ( B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C ) )
20	15, 19	sbie 2408	. . 3 |- ( [ y / x ] B R C <-> [_ y / x ]_ B [_ y / x ]_ R [_ y / x ]_ C )
21	7, 11, 20	vtoclbg 3267	. 2 |- ( A e. _V -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
22	1, 6, 21	pm5.21nii 368	1 |- ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  sbcbr  4707  sbcbr12g  4708  csbcnvgALT  5307  sbcfung  5912  csbfv12  6231  relowlpssretop  33212