Theorem sbcbr 4707

Description: Move substitution in and out of a binary relation. (Contributed by NM, 23-Aug-2018.)

Assertion

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

Distinct variable groups:   x, B   x, C

Allowed substitution hints:   A( x)   R( x)

Proof of Theorem sbcbr

Step	Hyp	Ref	Expression
1		sbcbr123 4706	. 2 |- ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C )
2		csbconstg 3546	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = B )
3		csbconstg 3546	. . . 4 |- ( A e. _V -> [_ A / x ]_ C = C )
4	2, 3	breq12d 4666	. . 3 |- ( A e. _V -> ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C <-> B [_ A / x ]_ R C ) )
5		br0 4701	. . . . 5 |- -. [_ A / x ]_ B (/) [_ A / x ]_ C
6		csbprc 3980	. . . . . 6 |- ( -. A e. _V -> [_ A / x ]_ R = (/) )
7	6	breqd 4664	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C <-> [_ A / x ]_ B (/) [_ A / x ]_ C ) )
8	5, 7	mtbiri 317	. . . 4 |- ( -. A e. _V -> -. [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C )
9		br0 4701	. . . . 5 |- -. B (/) C
10	6	breqd 4664	. . . . 5 |- ( -. A e. _V -> ( B [_ A / x ]_ R C <-> B (/) C ) )
11	9, 10	mtbiri 317	. . . 4 |- ( -. A e. _V -> -. B [_ A / x ]_ R C )
12	8, 11	2falsed 366	. . 3 |- ( -. A e. _V -> ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C <-> B [_ A / x ]_ R C ) )
13	4, 12	pm2.61i 176	. 2 |- ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C <-> B [_ A / x ]_ R C )
14	1, 13	bitri 264	1 |- ( [. A / x ]. B R C <-> B [_ A / x ]_ R C )

Syntax hints:   -. wn 3   <-> wb 196   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:  csbcnv  5306