Theorem csbov2g 5488

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 5486	. 2 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(B𝐹𝐶) = (⦋A / x⦌B𝐹⦋A / x⦌𝐶))
2		csbconstg 2858	. . 3 ⊢ (A ∈ 𝑉 → ⦋A / x⦌B = B)
3	2	oveq1d 5470	. 2 ⊢ (A ∈ 𝑉 → (⦋A / x⦌B𝐹⦋A / x⦌𝐶) = (B𝐹⦋A / x⦌𝐶))
4	1, 3	eqtrd 2069	1 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(B𝐹𝐶) = (B𝐹⦋A / x⦌𝐶))

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ⦋csb 2846  (class class class)co 5455

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458

This theorem is referenced by:  csbnegg  6966