Theorem csbov2g 7321

Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Assertion

Ref	Expression
csbov2g	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbov2g

Step	Hyp	Ref	Expression
1		csbov12g 7319	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))
2		csbconstg 3851	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
3	2	oveq1d 7290	. 2 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))
4	1, 3	eqtrd 2778	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ⦋csb 3832  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-dm 5599  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  csbnegg  11218  prmgaplem7  16758  matgsum  21586  scmatscm  21662  pm2mpf1lem  21943  pm2mpcoe1  21949  pm2mpmhmlem2  21968  monmat2matmon  21973  divcncf  24611  logbmpt  25938  finxpreclem4  35565  cotrclrcl  41350  ply1mulgsumlem3  45729  ply1mulgsumlem4  45730  ply1mulgsum  45731