Theorem csbiebt 3173

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3177.) (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 2868	. 2 ⊢ (A ∈ V → A ∈ V)
2		spsbc 3059	. . . . 5 ⊢ (A ∈ V → (∀x(x = A → B = C) → [̣A / x]̣(x = A → B = C)))
3	2	adantr 451	. . . 4 ⊢ ((A ∈ V ∧ ℲxC) → (∀x(x = A → B = C) → [̣A / x]̣(x = A → B = C)))
4		simpl 443	. . . . 5 ⊢ ((A ∈ V ∧ ℲxC) → A ∈ V)
5		biimt 325	. . . . . . 7 ⊢ (x = A → (B = C ↔ (x = A → B = C)))
6		csbeq1a 3145	. . . . . . . 8 ⊢ (x = A → B = [A / x]B)
7	6	eqeq1d 2361	. . . . . . 7 ⊢ (x = A → (B = C ↔ [A / x]B = C))
8	5, 7	bitr3d 246	. . . . . 6 ⊢ (x = A → ((x = A → B = C) ↔ [A / x]B = C))
9	8	adantl 452	. . . . 5 ⊢ (((A ∈ V ∧ ℲxC) ∧ x = A) → ((x = A → B = C) ↔ [A / x]B = C))
10		nfv 1619	. . . . . 6 ⊢ Ⅎx A ∈ V
11		nfnfc1 2493	. . . . . 6 ⊢ ℲxℲxC
12	10, 11	nfan 1824	. . . . 5 ⊢ Ⅎx(A ∈ V ∧ ℲxC)
13		nfcsb1v 3169	. . . . . . 7 ⊢ Ⅎx[A / x]B
14	13	a1i 10	. . . . . 6 ⊢ ((A ∈ V ∧ ℲxC) → Ⅎx[A / x]B)
15		simpr 447	. . . . . 6 ⊢ ((A ∈ V ∧ ℲxC) → ℲxC)
16	14, 15	nfeqd 2504	. . . . 5 ⊢ ((A ∈ V ∧ ℲxC) → Ⅎx[A / x]B = C)
17	4, 9, 12, 16	sbciedf 3082	. . . 4 ⊢ ((A ∈ V ∧ ℲxC) → ([̣A / x]̣(x = A → B = C) ↔ [A / x]B = C))
18	3, 17	sylibd 205	. . 3 ⊢ ((A ∈ V ∧ ℲxC) → (∀x(x = A → B = C) → [A / x]B = C))
19	13	a1i 10	. . . . . . . 8 ⊢ (ℲxC → Ⅎx[A / x]B)
20		id 19	. . . . . . . 8 ⊢ (ℲxC → ℲxC)
21	19, 20	nfeqd 2504	. . . . . . 7 ⊢ (ℲxC → Ⅎx[A / x]B = C)
22	11, 21	nfan1 1881	. . . . . 6 ⊢ Ⅎx(ℲxC ∧ [A / x]B = C)
23	7	biimprcd 216	. . . . . . 7 ⊢ ([A / x]B = C → (x = A → B = C))
24	23	adantl 452	. . . . . 6 ⊢ ((ℲxC ∧ [A / x]B = C) → (x = A → B = C))
25	22, 24	alrimi 1765	. . . . 5 ⊢ ((ℲxC ∧ [A / x]B = C) → ∀x(x = A → B = C))
26	25	ex 423	. . . 4 ⊢ (ℲxC → ([A / x]B = C → ∀x(x = A → B = C)))
27	26	adantl 452	. . 3 ⊢ ((A ∈ V ∧ ℲxC) → ([A / x]B = C → ∀x(x = A → B = C)))
28	18, 27	impbid 183	. 2 ⊢ ((A ∈ V ∧ ℲxC) → (∀x(x = A → B = C) ↔ [A / x]B = C))
29	1, 28	sylan 457	1 ⊢ ((A ∈ V ∧ ℲxC) → (∀x(x = A → B = C) ↔ [A / x]B = C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  Vcvv 2860  [̣wsbc 3047  [csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbiedf  3174  csbieb  3175  csbiegf  3177