Theorem sbcbr2g 3394

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 3392	. 2 |- (A e. D -> ([A / x]BRC <-> [_A / x]_BR[_A / x]_C))
2		ax-17 1317	. . . 4 |- (y e. B -> A.x y e. B)
3	2	csbconstgf 2551	. . 3 |- (A e. D -> [_A / x]_B = B)
4	3	breq1d 3348	. 2 |- (A e. D -> ([_A / x]_BR[_A / x]_C <-> BR[_A / x]_C))
5	1, 4	bitrd 587	1 |- (A e. D -> ([A / x]BRC <-> BR[_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 163   e. wcel 1300  [wsbc 1534  [_csb 2540   class class class wbr 3338

This theorem is referenced by:  bnj33 12401  fsum00 15820

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339

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