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Theorem csbov 6586

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 6585	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)
2		csbconstg 3512	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
3		csbconstg 3512	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶)
4	2, 3	oveq12d 6567	. . 3 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶))
5		0fv 6137	. . . . 5 ⊢ (∅‘⟨𝐵, 𝐶⟩) = ∅
6		df-ov 6552	. . . . 5 ⊢ (𝐵∅𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7		df-ov 6552	. . . . . 6 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = (∅‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩)
8		0fv 6137	. . . . . 6 ⊢ (∅‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩) = ∅
9	7, 8	eqtri 2632	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = ∅
10	5, 6, 9	3eqtr4ri 2643	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = (𝐵∅𝐶)
11		csbprc 3932	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
12	11	oveqd 6566	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶))
13	11	oveqd 6566	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶) = (𝐵∅𝐶))
14	10, 12, 13	3eqtr4a 2670	. . 3 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶))
15	4, 14	pm2.61i 175	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)
16	1, 15	eqtri 2632	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⦋csb 3499 ∅c0 3874 ⟨cop 4131 ‘cfv 5804 (class class class)co 6549

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 ax-pow 4769

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552

This theorem is referenced by: mptcoe1matfsupp 20426 mp2pm2mplem4 20433

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