Theorem csbov 7298

Description: Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbov	⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem csbov

Step	Hyp	Ref	Expression
1		csbov123 7297	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)
2		csbconstg 3847	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
3		csbconstg 3847	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶)
4	2, 3	oveq12d 7273	. . 3 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶))
5		0fv 6795	. . . . 5 ⊢ (∅‘⟨𝐵, 𝐶⟩) = ∅
6		df-ov 7258	. . . . 5 ⊢ (𝐵∅𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7		0ov 7292	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = ∅
8	5, 6, 7	3eqtr4ri 2777	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = (𝐵∅𝐶)
9		csbprc 4337	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
10	9	oveqd 7272	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶))
11	9	oveqd 7272	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶) = (𝐵∅𝐶))
12	8, 10, 11	3eqtr4a 2805	. . 3 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶))
13	4, 12	pm2.61i 182	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)
14	1, 13	eqtri 2766	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⦋csb 3828  ∅c0 4253  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-dm 5590  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  mptcoe1matfsupp  21859  mp2pm2mplem4  21866