Theorem sbcbr12g 4037

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

Assertion

Ref	Expression
sbcbr12g	|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )

Distinct variable group:   x, R

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

Proof of Theorem sbcbr12g

Step	Hyp	Ref	Expression
1		sbcbrg 4036	. 2 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
2		csbconstg 3059	. . 3 |- ( A e. D -> [_ A / x ]_ R = R )
3	2	breqd 3993	. 2 |- ( A e. D -> ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
4	1, 3	bitrd 187	1 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2136   [.wsbc 2951   [_csb 3045   class class class wbr 3982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983

This theorem is referenced by:  sbcbr1g  4038  sbcbr2g  4039