Theorem sbcbr12g 3978

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 3977	. 2 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
2		csbconstg 3011	. . 3 |- ( A e. D -> [_ A / x ]_ R = R )
3	2	breqd 3935	. 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 1480   [.wsbc 2904   [_csb 2998   class class class wbr 3924

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925

This theorem is referenced by:  sbcbr1g  3979  sbcbr2g  3980