Theorem sbcbr12g 2663

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 2662	. 2 |- (A e. D -> ([A / x]BRC <-> [_A / x]_B[_A / x]_R[_A / x]_C))
2		ax-17 971	. . . 4 |- (y e. R -> A.x y e. R)
3	2	csbconstgf 2010	. . 3 |- (A e. D -> [_A / x]_R = R)
4		breq 2621	. . 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 528	1 |- (A e. D -> ([A / x]BRC <-> [_A / x]_BR[_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  [wsbc 1170  [_csb 2001   class class class wbr 2619

This theorem is referenced by:  sbcbr1g 2664  sbcbr2g 2665  fsumcmp 7040

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620

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