Theorem sbcbr2g 4710

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

Assertion

Ref	Expression
sbcbr2g	|- ( A e. V -> ( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) )

Distinct variable groups:   x, B   x, R

Allowed substitution hints:   A( x)   C( x)   V( x)

Proof of Theorem sbcbr2g

Step	Hyp	Ref	Expression
1		sbcbr12g 4708	. 2 |- ( A e. V -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
2		csbconstg 3546	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	breq1d 4663	. 2 |- ( A e. V -> ( [_ A / x ]_ B R [_ A / x ]_ C <-> B R [_ A / x ]_ C ) )
4	1, 3	bitrd 268	1 |- ( A e. V -> ( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   [.wsbc 3435   [_csb 3533   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:  prmgaplem7  15761  telgsums  18390  fvmptnn04if  20654  bnj110  30928  frege124d  38053  frege72  38229  frege91  38248  frege116  38273  frege120  38277