Theorem sbcbr12g 4044

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 4043	. 2 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
2		csbconstg 3063	. . 3 |- ( A e. D -> [_ A / x ]_ R = R )
3	2	breqd 4000	. 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 2141   [.wsbc 2955   [_csb 3049   class class class wbr 3989

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by:  sbcbr1g  4045  sbcbr2g  4046