Theorem sbcbr2g 4100

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 4098	. 2 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
2		csbconstg 3106	. . 3 |- ( A e. D -> [_ A / x ]_ B = B )
3	2	breq1d 4053	. 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 2175   [.wsbc 2997   [_csb 3092   class class class wbr 4043

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044

This theorem is referenced by: (None)