Theorem csbov1g 5705

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

Assertion

Ref	Expression
csbov1g	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹𝐶))

Distinct variable groups:   𝑥,𝐶   𝑥,𝐹

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

Proof of Theorem csbov1g

Step	Hyp	Ref	Expression
1		csbov12g 5704	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))
2		csbconstg 2948	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶)
3	2	oveq2d 5684	. 2 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹𝐶))
4	1, 3	eqtrd 2121	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹𝐶))

Syntax hints:   → wi 4   = wceq 1290   ∈ wcel 1439  ⦋csb 2936  (class class class)co 5668

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2624  df-sbc 2844  df-csb 2937  df-un 3006  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-iota 4995  df-fv 5038  df-ov 5671

This theorem is referenced by:  modfsummodlemstep  10914