Theorem sbcbrg 2662

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

Assertion

Ref	Expression
sbcbrg	|- (A e. D -> ([A / x]BRC <-> [_A / x]_B[_A / x]_R[_A / x]_C))

Proof of Theorem sbcbrg

Step	Hyp	Ref	Expression
1		ax-17 971	. . . 4 |- (A e. D -> A.y A e. D)
2		ax-17 971	. . . . 5 |- (z e. A -> A.y z e. A)
3	2	hbcsb1g 2024	. . . 4 |- (A e. D -> (z e. [_A / y]_[_y / x]_B -> A.y z e. [_A / y]_[_y / x]_B))
4	2	hbcsb1g 2024	. . . 4 |- (A e. D -> (z e. [_A / y]_[_y / x]_R -> A.y z e. [_A / y]_[_y / x]_R))
5	2	hbcsb1g 2024	. . . 4 |- (A e. D -> (z e. [_A / y]_[_y / x]_C -> A.y z e. [_A / y]_[_y / x]_C))
6	1, 3, 4, 5	hbbrd 2659	. . 3 |- (A e. D -> ([_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C -> A.y[_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C))
7		a9e 1125	. . . . . 6 |- E.x x = y
8		visset 1813	. . . . . . . . 9 |- y e. V
9		ax-17 971	. . . . . . . . . 10 |- (z e. y -> A.x z e. y)
10	9	hbsbc1g 1948	. . . . . . . . 9 |- (y e. V -> ([y / x]BRC -> A.x[y / x]BRC))
11	8, 10	ax-mp 7	. . . . . . . 8 |- ([y / x]BRC -> A.x[y / x]BRC)
12	8, 9	hbcsb1 2025	. . . . . . . . 9 |- (z e. [_y / x]_B -> A.x z e. [_y / x]_B)
13	8, 9	hbcsb1 2025	. . . . . . . . 9 |- (z e. [_y / x]_R -> A.x z e. [_y / x]_R)
14	8, 9	hbcsb1 2025	. . . . . . . . 9 |- (z e. [_y / x]_C -> A.x z e. [_y / x]_C)
15	12, 13, 14	hbbr 2658	. . . . . . . 8 |- ([_y / x]_B[_y / x]_R[_y / x]_C -> A.x[_y / x]_B[_y / x]_R[_y / x]_C)
16	11, 15	hbbi 1010	. . . . . . 7 |- (([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C) -> A.x([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C))
17		csbeq1a 2006	. . . . . . . . 9 |- (x = y -> B = [_y / x]_B)
18		csbeq1a 2006	. . . . . . . . 9 |- (x = y -> C = [_y / x]_C)
19	17, 18	breq12d 2631	. . . . . . . 8 |- (x = y -> (BRC <-> [_y / x]_BR[_y / x]_C))
20		sbceq1a 1944	. . . . . . . 8 |- (x = y -> (BRC <-> [y / x]BRC))
21		csbeq1a 2006	. . . . . . . . 9 |- (x = y -> R = [_y / x]_R)
22		breq 2621	. . . . . . . . 9 |- (R = [_y / x]_R -> ([_y / x]_BR[_y / x]_C <-> [_y / x]_B[_y / x]_R[_y / x]_C))
23	21, 22	syl 10	. . . . . . . 8 |- (x = y -> ([_y / x]_BR[_y / x]_C <-> [_y / x]_B[_y / x]_R[_y / x]_C))
24	19, 20, 23	3bitr3d 548	. . . . . . 7 |- (x = y -> ([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C))
25	16, 24	19.23ai 1064	. . . . . 6 |- (E.x x = y -> ([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C))
26	7, 25	ax-mp 7	. . . . 5 |- ([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C)
27	26	a1i 8	. . . 4 |- (y = A -> ([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C))
28		csbeq1a 2006	. . . . 5 |- (y = A -> [_y / x]_B = [_A / y]_[_y / x]_B)
29		csbeq1a 2006	. . . . 5 |- (y = A -> [_y / x]_C = [_A / y]_[_y / x]_C)
30	28, 29	breq12d 2631	. . . 4 |- (y = A -> ([_y / x]_B[_y / x]_R[_y / x]_C <-> [_A / y]_[_y / x]_B[_y / x]_R[_A / y]_[_y / x]_C))
31		csbeq1a 2006	. . . . 5 |- (y = A -> [_y / x]_R = [_A / y]_[_y / x]_R)
32		breq 2621	. . . . 5 |- ([_y / x]_R = [_A / y]_[_y / x]_R -> ([_A / y]_[_y / x]_B[_y / x]_R[_A / y]_[_y / x]_C <-> [_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C))
33	31, 32	syl 10	. . . 4 |- (y = A -> ([_A / y]_[_y / x]_B[_y / x]_R[_A / y]_[_y / x]_C <-> [_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C))
34	27, 30, 33	3bitrd 544	. . 3 |- (y = A -> ([y / x]BRC <-> [_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C))
35	6, 34	sbciegf 1960	. 2 |- (A e. D -> ([A / y][y / x]BRC <-> [_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C))
36		sbccog 1952	. 2 |- (A e. D -> ([A / y][y / x]BRC <-> [A / x]BRC))
37		csbcog 2007	. . . 4 |- (A e. D -> [_A / y]_[_y / x]_B = [_A / x]_B)
38		csbcog 2007	. . . 4 |- (A e. D -> [_A / y]_[_y / x]_C = [_A / x]_C)
39	37, 38	breq12d 2631	. . 3 |- (A e. D -> ([_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C <-> [_A / x]_B[_A / y]_[_y / x]_R[_A / x]_C))
40		csbcog 2007	. . . 4 |- (A e. D -> [_A / y]_[_y / x]_R = [_A / x]_R)
41		breq 2621	. . . 4 |- ([_A / y]_[_y / x]_R = [_A / x]_R -> ([_A / x]_B[_A / y]_[_y / x]_R[_A / x]_C <-> [_A / x]_B[_A / x]_R[_A / x]_C))
42	40, 41	syl 10	. . 3 |- (A e. D -> ([_A / x]_B[_A / y]_[_y / x]_R[_A / x]_C <-> [_A / x]_B[_A / x]_R[_A / x]_C))
43	39, 42	bitrd 528	. 2 |- (A e. D -> ([_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C <-> [_A / x]_B[_A / x]_R[_A / x]_C))
44	35, 36, 43	3bitr3d 548	1 |- (A e. D -> ([A / x]BRC <-> [_A / x]_B[_A / x]_R[_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  Vcvv 1811  [_csb 2001   class class class wbr 2619

This theorem is referenced by:  sbcbr12g 2663

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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