Theorem sbcbrgOLD 3391

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

Assertion

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

Proof of Theorem sbcbrgOLD

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . 4 |- (A e. D -> A.y A e. D)
2		ax-17 1317	. . . . 5 |- (z e. A -> A.y z e. A)
3	2	hbcsb1g 2567	. . . 4 |- (A e. D -> (z e. [_A / y]_[_y / x]_B -> A.y z e. [_A / y]_[_y / x]_B))
4	2	hbcsb1g 2567	. . . 4 |- (A e. D -> (z e. [_A / y]_[_y / x]_R -> A.y z e. [_A / y]_[_y / x]_R))
5	2	hbcsb1g 2567	. . . 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 3382	. . 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 1483	. . . . . 6 |- E.x x = y
8		visset 2295	. . . . . . . . 9 |- y e. _V
9		ax-17 1317	. . . . . . . . . 10 |- (z e. y -> A.x z e. y)
10	9	hbsbc1g 2461	. . . . . . . . 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 2568	. . . . . . . . 9 |- (z e. [_y / x]_B -> A.x z e. [_y / x]_B)
13	8, 9	hbcsb1 2568	. . . . . . . . 9 |- (z e. [_y / x]_R -> A.x z e. [_y / x]_R)
14	8, 9	hbcsb1 2568	. . . . . . . . 9 |- (z e. [_y / x]_C -> A.x z e. [_y / x]_C)
15	12, 13, 14	hbbr 3381	. . . . . . . 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 1357	. . . . . . 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 2546	. . . . . . . . 9 |- (x = y -> B = [_y / x]_B)
18		csbeq1a 2546	. . . . . . . . 9 |- (x = y -> C = [_y / x]_C)
19	17, 18	breq12d 3351	. . . . . . . 8 |- (x = y -> (BRC <-> [_y / x]_BR[_y / x]_C))
20		sbceq1a 2456	. . . . . . . 8 |- (x = y -> (BRC <-> [y / x]BRC))
21		csbeq1a 2546	. . . . . . . . 9 |- (x = y -> R = [_y / x]_R)
22	21	breqd 3349	. . . . . . . 8 |- (x = y -> ([_y / x]_BR[_y / x]_C <-> [_y / x]_B[_y / x]_R[_y / x]_C))
23	19, 20, 22	3bitr3d 607	. . . . . . 7 |- (x = y -> ([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C))
24	16, 23	19.23ai 1412	. . . . . 6 |- (E.x x = y -> ([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C))
25	7, 24	ax-mp 7	. . . . 5 |- ([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C)
26	25	a1i 8	. . . 4 |- (y = A -> ([y / x]BRC <-> [_y / x]_B[_y / x]_R[_y / x]_C))
27		csbeq1a 2546	. . . . 5 |- (y = A -> [_y / x]_B = [_A / y]_[_y / x]_B)
28		csbeq1a 2546	. . . . 5 |- (y = A -> [_y / x]_C = [_A / y]_[_y / x]_C)
29	27, 28	breq12d 3351	. . . 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))
30		csbeq1a 2546	. . . . 5 |- (y = A -> [_y / x]_R = [_A / y]_[_y / x]_R)
31	30	breqd 3349	. . . 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))
32	26, 29, 31	3bitrd 603	. . 3 |- (y = A -> ([y / x]BRC <-> [_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C))
33	6, 32	sbciegf 2483	. 2 |- (A e. D -> ([A / y][y / x]BRC <-> [_A / y]_[_y / x]_B[_A / y]_[_y / x]_R[_A / y]_[_y / x]_C))
34		sbccog 2467	. 2 |- (A e. D -> ([A / y][y / x]BRC <-> [A / x]BRC))
35		csbcog 2547	. . 3 |- (A e. D -> [_A / y]_[_y / x]_B = [_A / x]_B)
36		csbcog 2547	. . 3 |- (A e. D -> [_A / y]_[_y / x]_R = [_A / x]_R)
37		csbcog 2547	. . 3 |- (A e. D -> [_A / y]_[_y / x]_C = [_A / x]_C)
38	35, 36, 37	breq123d 3353	. 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))
39	33, 34, 38	3bitr3d 607	1 |- (A e. D -> ([A / x]BRC <-> [_A / x]_B[_A / x]_R[_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  _Vcvv 2292  [_csb 2540   class class class wbr 3338

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-3an 860  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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