Theorem csboprg 3981

Description: Move class substitution in and out of an operation.

Assertion

Ref	Expression
csboprg	⊢ (A ∈ D → [A / x](BFC) = ([A / x]B[A / x]F[A / x]C))

Proof of Theorem csboprg

Step	Hyp	Ref	Expression
1		ax-17 970	. . . 4 ⊢ (A ∈ D → ∀y A ∈ D)
2		ax-17 970	. . . . 5 ⊢ (z ∈ A → ∀y z ∈ A)
3	2	hbcsb1g 2021	. . . 4 ⊢ (A ∈ D → (z ∈ [A / y][y / x]B → ∀y z ∈ [A / y][y / x]B))
4	2	hbcsb1g 2021	. . . 4 ⊢ (A ∈ D → (z ∈ [A / y][y / x]F → ∀y z ∈ [A / y][y / x]F))
5	2	hbcsb1g 2021	. . . 4 ⊢ (A ∈ D → (z ∈ [A / y][y / x]C → ∀y z ∈ [A / y][y / x]C))
6	1, 3, 4, 5	hboprd 3977	. . 3 ⊢ (A ∈ D → (z ∈ ([A / y][y / x]B[A / y][y / x]F[A / y][y / x]C) → ∀y z ∈ ([A / y][y / x]B[A / y][y / x]F[A / y][y / x]C)))
7		a9e 1124	. . . . . 6 ⊢ ∃x x = y
8		visset 1810	. . . . . . . . 9 ⊢ y ∈ V
9		ax-17 970	. . . . . . . . 9 ⊢ (z ∈ y → ∀x z ∈ y)
10	8, 9	hbcsb1 2022	. . . . . . . 8 ⊢ (z ∈ [y / x](BFC) → ∀x z ∈ [y / x](BFC))
11	8, 9	hbcsb1 2022	. . . . . . . . 9 ⊢ (z ∈ [y / x]B → ∀x z ∈ [y / x]B)
12	8, 9	hbcsb1 2022	. . . . . . . . 9 ⊢ (z ∈ [y / x]F → ∀x z ∈ [y / x]F)
13	8, 9	hbcsb1 2022	. . . . . . . . 9 ⊢ (z ∈ [y / x]C → ∀x z ∈ [y / x]C)
14	11, 12, 13	hbopr 3976	. . . . . . . 8 ⊢ (z ∈ ([y / x]B[y / x]F[y / x]C) → ∀x z ∈ ([y / x]B[y / x]F[y / x]C))
15	10, 14	hbeq 1563	. . . . . . 7 ⊢ ([y / x](BFC) = ([y / x]B[y / x]F[y / x]C) → ∀x[y / x](BFC) = ([y / x]B[y / x]F[y / x]C))
16		csbeq1a 2003	. . . . . . . . 9 ⊢ (x = y → B = [y / x]B)
17		csbeq1a 2003	. . . . . . . . 9 ⊢ (x = y → C = [y / x]C)
18	16, 17	opreq12d 3973	. . . . . . . 8 ⊢ (x = y → (BFC) = ([y / x]BF[y / x]C))
19		csbeq1a 2003	. . . . . . . 8 ⊢ (x = y → (BFC) = [y / x](BFC))
20		csbeq1a 2003	. . . . . . . . 9 ⊢ (x = y → F = [y / x]F)
21	20	opreqd 3972	. . . . . . . 8 ⊢ (x = y → ([y / x]BF[y / x]C) = ([y / x]B[y / x]F[y / x]C))
22	18, 19, 21	3eqtr3d 1513	. . . . . . 7 ⊢ (x = y → [y / x](BFC) = ([y / x]B[y / x]F[y / x]C))
23	15, 22	19.23ai 1063	. . . . . 6 ⊢ (∃x x = y → [y / x](BFC) = ([y / x]B[y / x]F[y / x]C))
24	7, 23	ax-mp 7	. . . . 5 ⊢ [y / x](BFC) = ([y / x]B[y / x]F[y / x]C)
25	24	a1i 8	. . . 4 ⊢ (y = A → [y / x](BFC) = ([y / x]B[y / x]F[y / x]C))
26		csbeq1a 2003	. . . . 5 ⊢ (y = A → [y / x]B = [A / y][y / x]B)
27		csbeq1a 2003	. . . . 5 ⊢ (y = A → [y / x]C = [A / y][y / x]C)
28	26, 27	opreq12d 3973	. . . 4 ⊢ (y = A → ([y / x]B[y / x]F[y / x]C) = ([A / y][y / x]B[y / x]F[A / y][y / x]C))
29		csbeq1a 2003	. . . . 5 ⊢ (y = A → [y / x]F = [A / y][y / x]F)
30	29	opreqd 3972	. . . 4 ⊢ (y = A → ([A / y][y / x]B[y / x]F[A / y][y / x]C) = ([A / y][y / x]B[A / y][y / x]F[A / y][y / x]C))
31	25, 28, 30	3eqtrd 1509	. . 3 ⊢ (y = A → [y / x](BFC) = ([A / y][y / x]B[A / y][y / x]F[A / y][y / x]C))
32	6, 31	csbiegf 2028	. 2 ⊢ (A ∈ D → [A / y][y / x](BFC) = ([A / y][y / x]B[A / y][y / x]F[A / y][y / x]C))
33		csbcog 2004	. 2 ⊢ (A ∈ D → [A / y][y / x](BFC) = [A / x](BFC))
34		csbcog 2004	. . . 4 ⊢ (A ∈ D → [A / y][y / x]B = [A / x]B)
35		csbcog 2004	. . . 4 ⊢ (A ∈ D → [A / y][y / x]C = [A / x]C)
36	34, 35	opreq12d 3973	. . 3 ⊢ (A ∈ D → ([A / y][y / x]B[A / y][y / x]F[A / y][y / x]C) = ([A / x]B[A / y][y / x]F[A / x]C))
37		csbcog 2004	. . . 4 ⊢ (A ∈ D → [A / y][y / x]F = [A / x]F)
38	37	opreqd 3972	. . 3 ⊢ (A ∈ D → ([A / x]B[A / y][y / x]F[A / x]C) = ([A / x]B[A / x]F[A / x]C))
39	36, 38	eqtrd 1505	. 2 ⊢ (A ∈ D → ([A / y][y / x]B[A / y][y / x]F[A / y][y / x]C) = ([A / x]B[A / x]F[A / x]C))
40	32, 33, 39	3eqtr3d 1513	1 ⊢ (A ∈ D → [A / x](BFC) = ([A / x]B[A / x]F[A / x]C))

Syntax hints:   → wi 3   = wceq 955   ∈ wcel 957  ∃wex 979  [csb 1998  (class class class)co 3958

This theorem is referenced by:  csbopr12g 3982

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-13 968  ax-14 969  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  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194  df-opr 3960

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