Theorem csbov2g

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

	Ref	Expression
Assertion	csbov2g	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐵 𝐹 𝐶 ) = ( 𝐵 𝐹 ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		csbov12g	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐵 𝐹 𝐶 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐵 𝐹 ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) )
2		csbconstg	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐵 )
3	2	oveq1d	⊢ ( 𝐴 ∈ 𝑉 → ( ⦋ 𝐴 / 𝑥 ⦌ 𝐵 𝐹 ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) = ( 𝐵 𝐹 ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) )
4	1 3	eqtrd	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐵 𝐹 𝐶 ) = ( 𝐵 𝐹 ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) )