Theorem csbaovg 44623

Description: Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

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

Proof of Theorem csbaovg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3839	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌ ((𝐵𝐹𝐶)) = ⦋𝐴 / 𝑥⦌ ((𝐵𝐹𝐶)) )
2		csbeq1 3839	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3839	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3839	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	aoveq123d 44621	. . 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 2755	. 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 3434	. . 3 ⊢ 𝑦 ∈ V
8		nfcsb1v 3861	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcsb1v 3861	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
10		nfcsb1v 3861	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfaov 44622	. . 3 ⊢ Ⅎ𝑥 ((⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶))
12		csbeq1a 3850	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
13		csbeq1a 3850	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3850	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	aoveq123d 44621	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵𝐹𝐶)) = ((⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶)) )
16	7, 11, 15	csbief 3871	. 2 ⊢ ⦋𝑦 / 𝑥⦌ ((𝐵𝐹𝐶)) = ((⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶))
17	6, 16	vtoclg 3503	1 ⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌ ((𝐵𝐹𝐶)) = ((⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)) )

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2109  ⦋csb 3836   ((caov 44561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-int 4885  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-res 5600  df-iota 6388  df-fun 6432  df-fv 6438  df-aiota 44528  df-dfat 44562  df-afv 44563  df-aov 44564

This theorem is referenced by: (None)