Theorem sbcbr12g 4099

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 4098	. 2 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
2		csbconstg 3107	. . 3 |- ( A e. D -> [_ A / x ]_ R = R )
3	2	breqd 4055	. 2 |- ( A e. D -> ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
4	1, 3	bitrd 188	1 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2176   [.wsbc 2998   [_csb 3093   class class class wbr 4044

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045

This theorem is referenced by:  sbcbr1g  4100  sbcbr2g  4101