Theorem csbovg 5552

Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Assertion

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

Proof of Theorem csbovg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3139	. . 3 ⊢ (y = A → [y / x](BFC) = [A / x](BFC))
2		csbeq1 3139	. . . 4 ⊢ (y = A → [y / x]F = [A / x]F)
3		csbeq1 3139	. . . 4 ⊢ (y = A → [y / x]B = [A / x]B)
4		csbeq1 3139	. . . 4 ⊢ (y = A → [y / x]C = [A / x]C)
5	2, 3, 4	oveq123d 5543	. . 3 ⊢ (y = A → ([y / x]B[y / x]F[y / x]C) = ([A / x]B[A / x]F[A / x]C))
6	1, 5	eqeq12d 2367	. 2 ⊢ (y = A → ([y / x](BFC) = ([y / x]B[y / x]F[y / x]C) ↔ [A / x](BFC) = ([A / x]B[A / x]F[A / x]C)))
7		vex 2862	. . 3 ⊢ y ∈ V
8		nfcsb1v 3168	. . . 4 ⊢ Ⅎx[y / x]B
9		nfcsb1v 3168	. . . 4 ⊢ Ⅎx[y / x]F
10		nfcsb1v 3168	. . . 4 ⊢ Ⅎx[y / x]C
11	8, 9, 10	nfov 5545	. . 3 ⊢ Ⅎx([y / x]B[y / x]F[y / x]C)
12		csbeq1a 3144	. . . 4 ⊢ (x = y → F = [y / x]F)
13		csbeq1a 3144	. . . 4 ⊢ (x = y → B = [y / x]B)
14		csbeq1a 3144	. . . 4 ⊢ (x = y → C = [y / x]C)
15	12, 13, 14	oveq123d 5543	. . 3 ⊢ (x = y → (BFC) = ([y / x]B[y / x]F[y / x]C))
16	7, 11, 15	csbief 3177	. 2 ⊢ [y / x](BFC) = ([y / x]B[y / x]F[y / x]C)
17	6, 16	vtoclg 2914	1 ⊢ (A ∈ D → [A / x](BFC) = ([A / x]B[A / x]F[A / x]C))

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  [csb 3136  (class class class)co 5525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795  df-ov 5526

This theorem is referenced by:  csbov12g  5553