Theorem csbiebt 3088

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3092.) (Contributed by NM, 11-Nov-2005.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiebt

Step	Hyp	Ref	Expression
1		elex 2741	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		spsbc 2966	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
3	2	adantr 274	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
4		simpl 108	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → 𝐴 ∈ V)
5		biimt 240	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶)))
6		csbeq1a 3058	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
7	6	eqeq1d 2179	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 7	bitr3d 189	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
9	8	adantl 275	. . . . 5 ⊢ (((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
10		nfv 1521	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ∈ V
11		nfnfc1 2315	. . . . . 6 ⊢ Ⅎ𝑥Ⅎ𝑥𝐶
12	10, 11	nfan 1558	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ Ⅎ𝑥𝐶)
13		nfcsb1v 3082	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
14	13	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
15		simpr 109	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥𝐶)
16	14, 15	nfeqd 2327	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
17	4, 9, 12, 16	sbciedf 2990	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
18	3, 17	sylibd 148	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
19	13	a1i 9	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
20		id 19	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥𝐶)
21	19, 20	nfeqd 2327	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
22	11, 21	nfan1 1557	. . . . . 6 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
23	7	biimprcd 159	. . . . . . 7 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
24	23	adantl 275	. . . . . 6 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → (𝑥 = 𝐴 → 𝐵 = 𝐶))
25	22, 24	alrimi 1515	. . . . 5 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
26	25	ex 114	. . . 4 ⊢ (Ⅎ𝑥𝐶 → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
27	26	adantl 275	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
28	18, 27	impbid 128	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
29	1, 28	sylan 281	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348   ∈ wcel 2141  Ⅎwnfc 2299  Vcvv 2730  [wsbc 2955  ⦋csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  csbiedf  3089  csbieb  3090  csbiegf  3092