Theorem csbov 7496

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 7495	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)
2		csbconstg 3870	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
3		csbconstg 3870	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶)
4	2, 3	oveq12d 7469	. . 3 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶))
5		0fv 6957	. . . . 5 ⊢ (∅‘⟨𝐵, 𝐶⟩) = ∅
6		df-ov 7454	. . . . 5 ⊢ (𝐵∅𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7		0ov 7488	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = ∅
8	5, 6, 7	3eqtr4ri 2797	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶) = (𝐵∅𝐶)
9		csbprc 4370	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
10	9	oveqd 7468	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶))
11	9	oveqd 7468	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶) = (𝐵∅𝐶))
12	8, 10, 11	3eqtr4a 2824	. . 3 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶))
13	4, 12	pm2.61i 184	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)
14	1, 13	eqtri 2786	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⦋csb 3851  ∅c0 4281  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574  df-ov 7454

This theorem is used by:  mptcoe1matfsupp  23175  mp2pm2mplem4  23182