Theorem sbcbr12g 3814

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 3813	. 2 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
2		csbconstg 2864	. . 3 |- ( A e. D -> [_ A / x ]_ R = R )
3	2	breqd 3775	. 2 |- ( A e. D -> ( [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
4	1, 3	bitrd 177	1 |- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   [.wsbc 2764   [_csb 2852   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765

This theorem is referenced by:  sbcbr1g  3815  sbcbr2g  3816