Theorem csbiebt

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf .) (Contributed by NM, 11-Nov-2005)

	Ref	Expression
Assertion	csbiebt	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Ⅎ 𝑥 𝐶 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
2		spsbc	⊢ ( 𝐴 ∈ V → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) → [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ) )
3	2	adantr	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) → [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ) )
4		simpl	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → 𝐴 ∈ V )
5		biimt	⊢ ( 𝑥 = 𝐴 → ( 𝐵 = 𝐶 ↔ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ) )
6		csbeq1a	⊢ ( 𝑥 = 𝐴 → 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
7	6	eqeq1d	⊢ ( 𝑥 = 𝐴 → ( 𝐵 = 𝐶 ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) )
8	5 7	bitr3d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) )
9	8	adantl	⊢ ( ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) ∧ 𝑥 = 𝐴 ) → ( ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) )
10		nfv	⊢ Ⅎ 𝑥 𝐴 ∈ V
11		nfnfc1	⊢ Ⅎ 𝑥 Ⅎ 𝑥 𝐶
12	10 11	nfan	⊢ Ⅎ 𝑥 ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 )
13		nfcsb1v	⊢ Ⅎ 𝑥 ⦋ 𝐴 / 𝑥 ⦌ 𝐵
14	13	a1i	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → Ⅎ 𝑥 ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
15		simpr	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → Ⅎ 𝑥 𝐶 )
16	14 15	nfeqd	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → Ⅎ 𝑥 ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )
17	4 9 12 16	sbciedf	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → ( [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) )
18	3 17	sylibd	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) )
19	13	a1i	⊢ ( Ⅎ 𝑥 𝐶 → Ⅎ 𝑥 ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
20		id	⊢ ( Ⅎ 𝑥 𝐶 → Ⅎ 𝑥 𝐶 )
21	19 20	nfeqd	⊢ ( Ⅎ 𝑥 𝐶 → Ⅎ 𝑥 ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )
22	11 21	nfan1	⊢ Ⅎ 𝑥 ( Ⅎ 𝑥 𝐶 ∧ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )
23	7	biimprcd	⊢ ( ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 → ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) )
24	23	adantl	⊢ ( ( Ⅎ 𝑥 𝐶 ∧ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) → ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) )
25	22 24	alrimi	⊢ ( ( Ⅎ 𝑥 𝐶 ∧ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) )
26	25	ex	⊢ ( Ⅎ 𝑥 𝐶 → ( ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ) )
27	26	adantl	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → ( ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ) )
28	18 27	impbid	⊢ ( ( 𝐴 ∈ V ∧ Ⅎ 𝑥 𝐶 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) )
29	1 28	sylan	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Ⅎ 𝑥 𝐶 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 ) )