Theorem elrabf 2994

Description: Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)

Hypotheses

Ref	Expression
elrabf.1	⊢ ℲxA
elrabf.2	⊢ ℲxB
elrabf.3	⊢ Ⅎxψ
elrabf.4	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
elrabf	⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ ψ))

Proof of Theorem elrabf

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ {x ∈ B ∣ φ} → A ∈ V)
2		elex 2868	. . 3 ⊢ (A ∈ B → A ∈ V)
3	2	adantr 451	. 2 ⊢ ((A ∈ B ∧ ψ) → A ∈ V)
4		df-rab 2624	. . . 4 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)}
5	4	eleq2i 2417	. . 3 ⊢ (A ∈ {x ∈ B ∣ φ} ↔ A ∈ {x ∣ (x ∈ B ∧ φ)})
6		elrabf.1	. . . 4 ⊢ ℲxA
7		elrabf.2	. . . . . 6 ⊢ ℲxB
8	6, 7	nfel 2498	. . . . 5 ⊢ Ⅎx A ∈ B
9		elrabf.3	. . . . 5 ⊢ Ⅎxψ
10	8, 9	nfan 1824	. . . 4 ⊢ Ⅎx(A ∈ B ∧ ψ)
11		eleq1 2413	. . . . 5 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
12		elrabf.4	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
13	11, 12	anbi12d 691	. . . 4 ⊢ (x = A → ((x ∈ B ∧ φ) ↔ (A ∈ B ∧ ψ)))
14	6, 10, 13	elabgf 2984	. . 3 ⊢ (A ∈ V → (A ∈ {x ∣ (x ∈ B ∧ φ)} ↔ (A ∈ B ∧ ψ)))
15	5, 14	syl5bb 248	. 2 ⊢ (A ∈ V → (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ ψ)))
16	1, 3, 15	pm5.21nii 342	1 ⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  {crab 2619  Vcvv 2860

This proof depends on 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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2624  df-v 2862

This theorem is used by:  elrab  2995