Theorem csbiebt 3177

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3181.) (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 2824	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		spsbc 3053	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
3	2	adantr 276	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
4		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → 𝐴 ∈ V)
5		biimt 241	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶)))
6		csbeq1a 3146	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
7	6	eqeq1d 2241	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 7	bitr3d 190	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
9	8	adantl 277	. . . . 5 ⊢ (((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
10		nfv 1577	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ∈ V
11		nfnfc1 2387	. . . . . 6 ⊢ Ⅎ𝑥Ⅎ𝑥𝐶
12	10, 11	nfan 1614	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ Ⅎ𝑥𝐶)
13		nfcsb1v 3170	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
14	13	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
15		simpr 110	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥𝐶)
16	14, 15	nfeqd 2399	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
17	4, 9, 12, 16	sbciedf 3077	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
18	3, 17	sylibd 149	. . 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 2399	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
22	11, 21	nfan1 1613	. . . . . 6 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
23	7	biimprcd 160	. . . . . . 7 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
24	23	adantl 277	. . . . . 6 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → (𝑥 = 𝐴 → 𝐵 = 𝐶))
25	22, 24	alrimi 1571	. . . . 5 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
26	25	ex 115	. . . 4 ⊢ (Ⅎ𝑥𝐶 → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
27	26	adantl 277	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
28	18, 27	impbid 129	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
29	1, 28	sylan 283	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2203  Ⅎwnfc 2371  Vcvv 2812  [wsbc 3041  ⦋csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-sbc 3042  df-csb 3138

This theorem is referenced by:  csbiedf  3178  csbieb  3179  csbiegf  3181