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Theorem csbov 7397

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 7396	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)
2		csbconstg 3865	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
3		csbconstg 3865	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶)
4	2, 3	oveq12d 7370	. . 3 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶))
5		0fv 6869	. . . . 5 ⊢ (∅‘⟨𝐵, 𝐶⟩) = ∅
6		df-ov 7355	. . . . 5 ⊢ (𝐵∅𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7		0ov 7389	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = ∅
8	5, 6, 7	3eqtr4ri 2767	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = (𝐵∅𝐶)
9		csbprc 4358	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
10	9	oveqd 7369	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶))
11	9	oveqd 7369	. . . 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 1541 ∈ wcel 2113 Vcvv 3437 ⦋csb 3846 ∅c0 4282 ⟨cop 4581 ‘cfv 6486 (class class class)co 7352

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-nul 5246 ax-pr 5372

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-ss 3915 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-dm 5629 df-iota 6442 df-fv 6494 df-ov 7355

This theorem is referenced by: mptcoe1matfsupp 22718 mp2pm2mplem4 22725