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Theorem csbov 7463

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 7462	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)
2		csbconstg 3911	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
3		csbconstg 3911	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶)
4	2, 3	oveq12d 7438	. . 3 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶))
5		0fv 6941	. . . . 5 ⊢ (∅‘⟨𝐵, 𝐶⟩) = ∅
6		df-ov 7423	. . . . 5 ⊢ (𝐵∅𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7		0ov 7457	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = ∅
8	5, 6, 7	3eqtr4ri 2767	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = (𝐵∅𝐶)
9		csbprc 4407	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
10	9	oveqd 7437	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶))
11	9	oveqd 7437	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶) = (𝐵∅𝐶))
12	8, 10, 11	3eqtr4a 2794	. . 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 2756	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⦋csb 3892 ∅c0 4323 ⟨cop 4635 ‘cfv 6548 (class class class)co 7420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-nul 5306 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-dm 5688 df-iota 6500 df-fv 6556 df-ov 7423

This theorem is referenced by: mptcoe1matfsupp 22703 mp2pm2mplem4 22710