Theorem sbcbr2g 2660

Description: Move substitution in and out of a binary relation.

Assertion

Ref	Expression
sbcbr2g	|- (A e. D -> ([A / x]BRC <-> BR[_A / x]_C))

Distinct variable groups:   x,B   x,R

Proof of Theorem sbcbr2g

Step	Hyp	Ref	Expression
1		sbcbr12g 2658	. 2 |- (A e. D -> ([A / x]BRC <-> [_A / x]_BR[_A / x]_C))
2		ax-17 969	. . . 4 |- (y e. B -> A.x y e. B)
3	2	csbconstgf 2006	. . 3 |- (A e. D -> [_A / x]_B = B)
4	3	breq1d 2624	. 2 |- (A e. D -> ([_A / x]_BR[_A / x]_C <-> BR[_A / x]_C))
5	1, 4	bitrd 527	1 |- (A e. D -> ([A / x]BRC <-> BR[_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 956  [wsbc 1168  [_csb 1997   class class class wbr 2614

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998  df-un 2046  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615

Copyright terms: Public domain