Theorem sbcbr2g 2670

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 2668	. 2 |- (A e. D -> ([A / x]BRC <-> [_A / x]_BR[_A / x]_C))
2		ax-17 973	. . . 4 |- (y e. B -> A.x y e. B)
3	2	csbconstgf 2013	. . 3 |- (A e. D -> [_A / x]_B = B)
4	3	breq1d 2634	. 2 |- (A e. D -> ([_A / x]_BR[_A / x]_C <-> BR[_A / x]_C))
5	1, 4	bitrd 530	1 |- (A e. D -> ([A / x]BRC <-> BR[_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 960  [wsbc 1172  [_csb 2004   class class class wbr 2624

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945  df-csb 2005  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625

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