Theorem elrabf 2980

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 2833	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → 𝐴 ∈ V)
2		elex 2833	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3	2	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → 𝐴 ∈ V)
4		df-rab 2537	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
5	4	eleq2i 2305	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
6		elrabf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
7		elrabf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfel 2401	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
9		elrabf.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
10	8, 9	nfan 1618	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
11		eleq1 2301	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
12		elrabf.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
13	11, 12	anbi12d 477	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
14	6, 10, 13	elabgf 2968	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
15	5, 14	bitrid 192	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
16	1, 3, 15	pm5.21nii 716	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  Ⅎwnf 1513   ∈ wcel 2209  {cab 2224  Ⅎwnfc 2379  {crab 2532  Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823

This theorem is referenced by:  elrab  2982  invdisjrab  4119  frind  4492  rabxfrd  4610  infssuzcldc  10646  nnwosdc  12794