Theorem sbcbr2g 4062

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

Assertion

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

Distinct variable groups:   x, B   x, R

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

Proof of Theorem sbcbr2g

Step	Hyp	Ref	Expression
1		sbcbr12g 4060	. 2 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
2		csbconstg 3073	. . 3 |- ( A e. D -> [_ A / x ]_ B = B )
3	2	breq1d 4015	. 2 |- ( A e. D -> ( [_ A / x ]_ B R [_ A / x ]_ C <-> B R [_ A / x ]_ C ) )
4	1, 3	bitrd 188	1 |- ( A e. D -> ( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2148   [.wsbc 2964   [_csb 3059   class class class wbr 4005

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006

This theorem is referenced by: (None)