Theorem sbcbr1g 4116

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

Assertion

Ref	Expression
sbcbr1g	|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R C ) )

Distinct variable groups:   x, C   x, R

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

Proof of Theorem sbcbr1g

Step	Hyp	Ref	Expression
1		sbcbr12g 4115	. 2 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
2		csbconstg 3115	. . 3 |- ( A e. D -> [_ A / x ]_ C = C )
3	2	breq2d 4071	. 2 |- ( A e. D -> ( [_ A / x ]_ B R [_ A / x ]_ C <-> [_ A / x ]_ B R C ) )
4	1, 3	bitrd 188	1 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R C ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2178   [.wsbc 3005   [_csb 3101   class class class wbr 4059

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060

This theorem is referenced by: (None)