Theorem csbaovg 48343

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 3849	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌ ((𝐵𝐹𝐶)) = ⦋𝐴 / 𝑥⦌ ((𝐵𝐹𝐶)) )
2		csbeq1 3849	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3849	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3849	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	aoveq123d 48341	. . 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 2776	. 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 3454	. . 3 ⊢ 𝑦 ∈ V
8		nfcsb1v 3870	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcsb1v 3870	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
10		nfcsb1v 3870	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfaov 48342	. . 3 ⊢ Ⅎ𝑥 ((⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶))
12		csbeq1a 3860	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
13		csbeq1a 3860	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3860	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	aoveq123d 48341	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵𝐹𝐶)) = ((⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶)) )
16	7, 11, 15	csbief 3880	. 2 ⊢ ⦋𝑦 / 𝑥⦌ ((𝐵𝐹𝐶)) = ((⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶))
17	6, 16	vtoclg 3517	1 ⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌ ((𝐵𝐹𝐶)) = ((⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)) )

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ⦋csb 3846   ((caov 48281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6494  df-fun 6540  df-fv 6546  df-aiota 48248  df-dfat 48282  df-afv 48283  df-aov 48284

This theorem is used by: (None)