Theorem csbidm

Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008) (Revised by NM, 18-Aug-2018)

	Ref	Expression
Assertion	csbidm	⊢ ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵

Proof

Step	Hyp	Ref	Expression
1		csbnest1g	⊢ ( 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ ⦋ 𝐴 / 𝑥 ⦌ 𝐴 / 𝑥 ⦌ 𝐵 )
2		csbconstg	⊢ ( 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ 𝐴 = 𝐴 )
3	2	csbeq1d	⊢ ( 𝐴 ∈ V → ⦋ ⦋ 𝐴 / 𝑥 ⦌ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
4	1 3	eqtrd	⊢ ( 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
5		csbprc	⊢ ( ¬ 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ∅ )
6		csbprc	⊢ ( ¬ 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ∅ )
7	5 6	eqtr4d	⊢ ( ¬ 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
8	4 7	pm2.61i	⊢ ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵