Theorem csbafv212g

Description: Move class substitution in and out of a function value, analogous to csbfv12 , with a direct proof proposed by Mario Carneiro, analogous to csbov123 . (Contributed by AV, 4-Sep-2022)

	Ref	Expression
Assertion	csbafv212g	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 '''' 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		csbeq1	⊢ ( 𝑦 = 𝐴 → ⦋ 𝑦 / 𝑥 ⦌ ( 𝐹 '''' 𝐵 ) = ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 '''' 𝐵 ) )
2		csbeq1	⊢ ( 𝑦 = 𝐴 → ⦋ 𝑦 / 𝑥 ⦌ 𝐹 = ⦋ 𝐴 / 𝑥 ⦌ 𝐹 )
3		csbeq1	⊢ ( 𝑦 = 𝐴 → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
4	2 3	afv2eq12d	⊢ ( 𝑦 = 𝐴 → ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) )
5	1 4	eqeq12d	⊢ ( 𝑦 = 𝐴 → ( ⦋ 𝑦 / 𝑥 ⦌ ( 𝐹 '''' 𝐵 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 '''' 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) )
6		vex	⊢ 𝑦 ∈ V
7		nfcsb1v	⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐹
8		nfcsb1v	⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵
9	7 8	nfafv2	⊢ Ⅎ 𝑥 ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 )
10		csbeq1a	⊢ ( 𝑥 = 𝑦 → 𝐹 = ⦋ 𝑦 / 𝑥 ⦌ 𝐹 )
11		csbeq1a	⊢ ( 𝑥 = 𝑦 → 𝐵 = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 )
12	10 11	afv2eq12d	⊢ ( 𝑥 = 𝑦 → ( 𝐹 '''' 𝐵 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) )
13	6 9 12	csbief	⊢ ⦋ 𝑦 / 𝑥 ⦌ ( 𝐹 '''' 𝐵 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝑦 / 𝑥 ⦌ 𝐵 )
14	5 13	vtoclg	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 '''' 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 '''' ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) )