Theorem csbov123g 6114

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
csbov123g	⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem csbov123g

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3150	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵𝐹𝐶) = ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶))
2		csbeq1 3150	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3150	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3150	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	oveq123d 6096	. . 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 2253	. 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 2824	. . 3 ⊢ 𝑦 ∈ V
8		nfcsb1v 3180	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcsb1v 3180	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
10		nfcsb1v 3180	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfov 6105	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶)
12		csbeq1a 3156	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
13		csbeq1a 3156	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3156	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	oveq123d 6096	. . 3 ⊢ (𝑥 = 𝑦 → (𝐵𝐹𝐶) = (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶))
16	7, 11, 15	csbief 3192	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶)
17	6, 16	vtoclg 2883	1 ⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  ⦋csb 3147  (class class class)co 6075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by:  csbov12g  6115