Theorem sbcbr12g 2659

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

Assertion

Ref	Expression
sbcbr12g	|- (A e. D -> ([A / x]BRC <-> [_A / x]_BR[_A / x]_C))

Distinct variable group:   x,R

Proof of Theorem sbcbr12g

Step	Hyp	Ref	Expression
1		sbcbrg 2658	. 2 |- (A e. D -> ([A / x]BRC <-> [_A / x]_B[_A / x]_R[_A / x]_C))
2		ax-17 970	. . . 4 |- (y e. R -> A.x y e. R)
3	2	csbconstgf 2007	. . 3 |- (A e. D -> [_A / x]_R = R)
4		breq 2617	. . 3 |- ([_A / x]_R = R -> ([_A / x]_B[_A / x]_R[_A / x]_C <-> [_A / x]_BR[_A / x]_C))
5	3, 4	syl 10	. 2 |- (A e. D -> ([_A / x]_B[_A / x]_R[_A / x]_C <-> [_A / x]_BR[_A / x]_C))
6	1, 5	bitrd 527	1 |- (A e. D -> ([A / x]BRC <-> [_A / x]_BR[_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  [wsbc 1169  [_csb 1998   class class class wbr 2615

This theorem is referenced by:  sbcbr1g 2660  sbcbr2g 2661  fsumcmp 6993

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939  df-csb 1999  df-un 2047  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616

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